
Oass.OA-33. 



Book >*4~l 



PRESENTED BY 



Hk 



MATHEMATICS 

FOR PRACTICAL MEN : 

BEING " 

A COMMON-PLACE BOOK 



PRINCIPLES, THEOREMS, RULES, ATSD TABLES, 
IN VARIOUS DEPARTMENTS 



PURE AND MIXED MATHEMATICS, 

WITH THEIR APPLICATION; 

ESPECIALLY 

TO THE PURSUITS OF SURVEYORS, ARCHITECTS, MECHANICS, 
AND CIVIL ENGINEERS. 



WITH NUMEROUS ENGRAVINGS. 



BY 

OLINTHUS GREGORY, LL.D., F.R.A.S., 

Corresponding Associate of the Academy of Dijon ; Honorary Member of the Literary and Philosophical Society of 

New York ; of the New York Historical Society ; of the Literary and Philosophical, and the Antiquarian 

Societies of Newcastle-upon-Tyne j of the Cambridge Philosophical Society ; of the Institution 

of Civil Engineers, &c &c, and Professor of Mathematics in the Royal Military Academy. 



SECOND AMERICAN FROM THE SECOND LONDON EDITION, 
CORRECTED AND IMPROVED. 



1 Only let men awake, and fix their eyes, one while on the nature of things, another while 
on the application of them to the use and service of mankind."— Lord Bacon. 



PHILADELPHIA : 
E. L. CAREY AND A. HART, CHESNUT ST. 

AMD SOLD BY ALL THE PRINCIPAL BOOKSELLERS IN THE UNITED STATES. 

18 36. 



<K** 



\*' 



GIFT 
HON. DAVID J. LEWIS 



<&*- 1 ,<£? 



TO 

THOMAS TELFORD, Esq. F.R.S.E. 

PRESIDENT; 
AND TO THE VARIOUS 

OFFICERS AND MEMBERS 

OF THE 

INSTITUTION OF CIVIL ENGINEERS; 
THIS COMPENDIUM 

OF 

MATHEMATICS FOR PRACTICAL MEN, 

IS MOST RESPECTFULLY DEDICATED 
BY THEIR 

FAITHFUL AND OBLIGED SERYANT 

THE AUTHOR. 



PREFACE. 



The work now presented to the public had its origin in a desire 
which I felt to draw up an Essay on the principles and applications 
of the mechanical sciences, for the use of the younger members of the 
Institution of Civil Engineers. The eminent individuals who are 
deservedly regarded as the main pillars of that useful Institution, 
stand in need of no such instructions as are in my power to impart : 
but it seemed expedient to prepare an Essay, comprised within 
moderate limits, which might furnish scientific instruction for the 
many young men of ardour and enterprise who have of late years 
devoted themselves to the interesting and important profession, of 
whose members that Institution is principally constituted. My first 
design was to compose a paper which might be read at one or two 
of the meetings of that Society ; but, as often happens in such cases, 
the embryo thought has grown, during meditation, from an essay to a 
book : and what was first meant to be a very compendious selection 
of principles and rules, has, in its execution, assumed the appearance 
of a systematic analysis of principles, theorems, rules, and tables. 

Indeed, the circumstances in which the inhabitants of this country 
are now placed, with regard to the love and acquisition of knowledge, 
impelled me, almost unconsciously, to such an extension of my 
original plan, as sprung from a desire to contribute to the instruction of 
that numerous class, the practical mechanics of this country. Besides 
the early disadvantages under which many of them have laboured, 
there is another which results from the activity of their pursuits. 
Unable, therefore, to go through the details of an extensive systematic 
course, they must, for the most part, be satisfied with imperfect views 
of theories and principles, and take much upon trust : an evil, how- 
ever, which the establishment, of Societies, and the composition of 
treatises, with an express view to their benefit, will probably soon 
diminish. 

Lord Brougham, in his "Practical Observations upon the Education 
of the People" remarks that " a most essential service will be rendered 

A 2 v 






VI PREFACE. 

to the cause of knowledge, by him who shall devote his time to the 
composition of elementary treatises on the Mathematics, sufficiently 
clear, and yet sufficiently compendious, to exemplify the method of 
reasoning employed in that science, and to impart an accurate know- 
ledge of the most useful fundamental propositions, with their appli- 
cation to practical purposes ; and treatises upon Natural Philosophy, 
which may teach the great principles of physics, and their practical 
application, to readers who have but a general knowledge of mathe- 
matics, or who are even wholly ignorant of the science beyond the 
common rules of arithmetic." And again, " He who shall prepare a 
treatise simply and concisely unfolding the doctrines of Algebra, 
Geometry, and Mechanics, and adding examples calculated to strike 
the imagination of their connexion with other branches of knowledge, 
and with the arts of common life, may fairly claim a large share in 
that rich harvest of discovery and invention which must be reaped by 
the thousands of ingenious and active men, thus enabled to bend their 
faculties towards objects at once useful and sublime." 

I do not attempt to persuade myself that the present volume will 
be thought adequately to supply the desiderata to which the passages 
advert ; yet I could not but be gratified, after full two-thirds of it were 
written, to find that the views which guided me in its execution ac- 
corded so far with the judgment of an individual, distinguished as 
Lord Brougham was, in early life, for the elegance and profundity 
of his mathematical researches. 

With a view to the elementary instruction of those who have not 
previously studied mathematics, I have commenced with brief, but, 1 
hope, perspicuous, treatises on Arithmetic and Algebra; a competent 
acquaintance with both of these being necessary to ensure that ac- 
curacy in computation which every practical man ought to attain, 
and that ready comprehension of scientific theorems and formulae 
which becomes the key to the stores of higher knowledge. As no 
man sharpens his tool or his weapon, merely that it may be sharp, but 
that it may be the fitter for use ; so no thoughtful man learns arith- 
metic and algebra for the mere sake of knowing those branches of 
science, but that he may employ them ; and these being possessed as 
valuable pre-requisites, the course of an author is thereby facilitated : 
for then, while he endeavours to express even common matters so 
that the learned shall not be disgusted, he may so express the more 
abstract and difficult that the comparatively ignorant (and the mere 
knowledge of arithmetic and algebra is, in our times, comparative 
ignorance) may practically understand and apply them. 



PREFACE. Vll 

After the first 103 pages, the remaining matter is synoptical. The 
general topics of geometry, trigonometry, conic sections, curves, 
perspective, mensuration, statics, dynamics, hydrostatics, hydrody- 
namics, and pneumatics, are thus treated. The definitions and prin- 
ciples are exhibited in an orderly series; but investigations and 
demonstrations are only sparingly introduced. This portion of the 
work is akin in its nature to a syllabus of a Course of Lectures on the 
departments of science which it treats ; with this difference, however, 
occasioned by the leading object of the publication, that popular 
illustrations are more frequently introduced, practical applications 
incessantly borne in mind, and such tables as seemed best calculated 
to save the labour of architects, mechanics, and civil engineers, 
inserted under their appropriate heads. Of these latter, several have 
been collected from former treatises, &c, but not a few have been 
either computed or contributed expressly for this Common-place Book. 

In a work like this, it would be absurd to pretend to originality. 
The plan, arrangement, and execution, are my own ; but the materials 
have long been regarded, and rightly, as common property. It has 
been my aim to reduce them into the smallest possible space, con- 
sistently with my general object; but, wherever I have found the 
work in this respect prepared to my hands, I have transcribed it into 
the following pages, with the usual references to the sources from 
whence it was taken. They who are conversant with the best writers 
on subjects of mixed mathematics and natural philosophy, will know 
that Smeaton, Robison, Playfair, Young, Bu Buat, Leslie, Hachette, 
Bland, Tredgold, fyc. are authors who ought to be consulted, in the 
preparation of a volume like this. I hope it will appear that I have 
duly, yet, at the same time, honourably, availed myself of the ad- 
vantages which they supply. I have, also, made such selections 
from my own earlier publications as were obviously suitable to my 
present purpose ; but not so copiously, I trust, as to diminish the 
utility of those volumes, or to make me an unfair borrower even from 
myself. 

Besides our junior Civil Engineers, and the numerous Practical 
Mechanics who are anxious to store their minds with scientific facts 
and principles ; there are others to whom, I flatter myself, the follow- 
ing pages will be found useful. Teachers of mathematics and those 
departments of natural philosophy which are introduced into our 
more respectable seminaries, may probably find this volume to occupy 
a convenient intermediate station between the merely popular ex- 
hibitions of the truths of mechanics, hydrostatics, &c, and the larger 



Vlll 



PREFACE. 



treatises, in which the whole chain of inquiry and demonstration is 
carefully presented, link by link, and the successive portions firmly 
connected upon irrefragable principles. While students who have 
recently terminated a scientific course, whether in our universities 
or other institutions, public or private, may, I would fain believe, find 
in this Common-place Book an abridged repository of the most 
valuable principles and theorems, and of hints for their applications 
to practical purposes. 

The only performances with which I am acquainted, that bear any 
direct analogy to this, are Martin's Young Student's Memorial Booh, 
Jones's Synopsis Palmariorum Matheseos, and Brunton's Compen- 
dium of Mechanics ; the latter of which I had not seen until the 
present volume was nearly completed. The first and last mentioned 
of these are neat and meritorious productions ; but restricted in their 
utility by the narrow space into which they are compressed. The 
other, written by the father of the late Sir William Jones, is a truly 
elegant introduction to the principles of Mathematics, considering the 
time in which it was written (1706) ; but as it is altogether theoretical, 
and is, moreover, now becoming exceedingly scarce, it by no means 
supersedes the necessity, for such I have been induced to regard it, 
of a Compendium like that which I now offer to the public. 

In its execution I have aimed at no higher reputation than that of 
being perspicuous, correct, and useful ; and if I shall be so fortunate 
as to have succeeded in those points, I shall be perfectly satisfied. 

Olinthus Gregory. 

Royal Military Academy, 
Woolwich, October 1st, 1825. 



In this new edition I have corrected a few errors which had escaped 
my notice in the former impression. I have also made a few such 
additions and improvements as the lapse of time and the progress of 
discovery rendered desirable ; and such as will, I hope, give the work 
new claims on public approbation. 

July 1st, 1833. 



CONTENTS. 



ARITHMETIC. 

Page 

Definitions and notation -- 1 

Addition of whole numbers 4 

Subtraction of whole numbers 5 

Multiplication of whole numbers ------ 6 

Division of whole numbers 10 

Proof of the first four rules 12 

Tables of weights and measures 14 

French and English weights and measures compared - - 21 

Vulgar fractions 23 

Addition, subtraction, multiplication, and division of ditto - 26 

Decimal fractions 28 

Reduction of decimals 29 

Addition, subtraction, multiplication, and division of ditto - - 32 

Complex fractions used in the arts and commerce - - - 33 

Addition, subtraction, &c. of ditto 35 

Duodecimals 38 

Powers and roots, square, cube, and higher roots - - - 39 

Proportion, rule of three, compound proportion, &c. - - 44 
Properties of numbers -------50 

Numerical problem on the reduction of ratios - - - - 53 

ALGEBRA. 

Definitions and notation 58 

Addition, subtraction, multiplication, division - - - - 60 

Involution of monomials and polynomials - - - - 67 

Evolution --70 

Surds, reduction, addition, subtraction, multiplication, division, 

involution, evolution 72 

Simple equations, extermination, &c. 78 

Quadratic equations --86 

Equations in general - - 91 

Progressions, arithmetical and geometrical - - - - 94 

Principles of logarithms 99 

Computation and use of formulae 101 

2 ix 



X CONTENTS. 

GEOMETRY. 

Page 
Definitions - 104 

Angles, right lines, and their rectangles 105 

Triangles 106 

Quadrilaterals and polygons 109 

Circles, and inscribed and circumscribed figures - - - 112 

Planes and solids 119 

Practical geometry 123 

TRIGONOMETRY. 

Plane trigonometry 133 

Determination of the heights and distances of objects - - 143 
by approximate and mechanical methods - - 150 

CONIC SECTIONS. 

Definitions, properties of the ellipse 155 

Properties of the hyperbola and parabola - 161 

General application to architecture - - - - - 167 

CURVES USEFUL TO ARCHITECTS, &C. 

Conchoid, cissoid, cycloid, quadratrix, catenary - - - 168 
Table of relations, strains, &c. in catenarian curves, applicable 

to chain bridges - - - 179 

Professor Farish's isometrical perspective - - - - 181 

MENSURATION. 

Mensuration of superficies 195 

Mensuration of solids 196 

Approximate rules for both surfaces and solids - - - 198 

MECHANICS. 

Definitions ---*-- -., 205 

STATICS. 

Parallelogram of forces, and applications - 206 

Centre of gravity 209 

Mechanical powers, lever, wheel and axle, pulley, inclined plane, 

wedge, screw 214 

General application of the principles of statics to the equilibrium 

of structures 222 

Equilibrium of piers 222 

Pressure of earth against walls 224 



CONTENTS. XI 

Page 
Equilibrium of polygons, centering roofs, &c. - 226 

Stability of arches 230 

Models : — Smart's mathematical chain-bridge - 232 

DYNAMICS. 

Definitions - - - 235 

Uniform motions 236 

Motion uniformly accelerated 236 

on pulleys, inclined planes, &c. ----- 238 

Motions about a centre or axis 243 

Pendulum, simple and compound, centres of oscillation, percus- 
sion, and gyration 244 

Portable pendulum 249 

Gridiron and other compensation pendulums - 253 

Table of lengths and vibrations of pendulums - - - - 255 

Principles of rotation ------- 259 

Central forces, steam-engine governors, fly- wheels - - - 261 

Percussion or collision 265 

Principles of chronometers, escapements, &c. - 269 

Select mechanical experiments 278 

HYDROSTATICS. 

Definitions - 284 

Pressure of non-elastic fluids 285 

Illustrations and applications : hydrostatic paradox and bellows* 
Bramah's press, embankments, strength of pipes of oak or 

iron - 286 

Floating bodies, buoyancy, Farey's self-acting flood gate - 291 

Specific gravities, Coates's hydrostatic balance, tables of specific 

gravities and weights of various substances - 293 
Rules for weights of leaden pipes, and rims of cast-iron fly- 
wheels - 300 

HYDRODYNAMICS. 

Definitions : motion and effluence of fluids - 302 

Motion of water in conduit pipes and open canals, over weirs, 

&c, with various tables 304 

Table of rise of water occasioned by piers of bridges and other 

contractions 316 

Contrivances to measure the velocity of running waters - - 317 
Effects of the old London bridge on the tides, &c. - - - 321 
Watermills : undershot, overshot, breast, Barker's - - - 325 

PNEUMATICS. 

Equilibrium of air and elastic fluids 330 



Xll CONTENTS. 

Page 

Pumps, sucking, lifting, forcing, fire-engine, Clark's quicksilver 

pump, Archimedes's screw, spiral pump, hydraulic ram, &c. 333 

Wind and windmills, Smeaton, Coulomb, &c. ... 345 

Steam and steam-engines; Savery, Newcomen and Cawley, 
Watt, Woolf, Oliver Evans, Fenton and Co., Nuncarrow, 
&c. - ■ 353 

Useful tables and remarks on steam-engines, rail-roads, canals, 

and turnpike roads ------- 373 

ACTIVE AND PASSIVE STRENGTH. 

Active strength, or animal energy, as of men, horses, &c. - 384 

Schulze's experiments, Coulomb's, Bevan's, Morisot's, Regnier's, 

Hachette's, Tredgold's, &c. 386 

Passive strength, modulus of elasticity, &c. - - - - 402 
Cohesive strength, modulus of cohesion, results of Leslie, Duha- 

mel, Rennie, Bevan, &c. 406 

SUPPLEMENTARY TABLES. 

Of useful factors connected with the circle - - - - 413 
Of circles ; from which knowing the diameters, the areas, cir- 
cumferences, and sides of equal squares are found - - 415 
Of relations of the arc, abscissa, ordinate, and subnormal, in the 

catenary 424 



PLATES. 

(To be placed at the end.) 

1. Isometrical Perspective. 

2. Steam-engines, Evans's, &c. 

3. Fenton and Co.'s engine. 



COMMON-PLACE BOOK, 

&c. &c. 



CHAPTER I. 

ARITHMETIC. 

Section I. — Definitions and Notation. 

Arithmetic is the science of numbers. 

We give the name of number to the assemblage of many 
units, or of many parts of an assumed unit ; unit being the 
quantity which, among all those of the same kind, forms a 
tvhole, which may be regarded as the base or element. Thus, 
when I speak of one house, one guinea, I speak of units, of 
which the first is the thing called a house, the second that 
called a guinea. But when I say four houses, ten guineas, 
three quarters of a guinea, I speak of numbers, of which the 
first is the unit house repeated four times ; the second is the 
unit guinea repeated ten times ; the third is the fourth part 
of the unit guinea repeated three times. 

In every particular classification of numbers, the unit is a 
measure taken arbitrarily, or established by usage and con- 
vention. 

Numbers formed by the repetition of an unbroken unit are 
called whole numbers, or integers, as seven miles, thirty shil- 
lings : those which are formed by the assemblage of many parts 
of a unit are called fractional numbers, or simply fractions ; 
as two-thirds of a yard, three-eighths of a mile. 

When the unit is restricted to a certain thing in particular, 
as one man, one horse, one pound, the collection of many of 
those units is called a concrete number, as ten men, tiventy 
horses, fifty pounds. But if the unit does not denote any parti- 
cular thing, and is expressed simply by one, numbers which are 
constituted of such units are denominated discrete or abstract, 
as five, ten, thirty. Hence, it is evident that abstract numbers 

B 1 



2 ARITHMETIC : NOTATION. 

can only be compared with their unit, as concrete numbers are 
compared with, or measured by, theirs ; but that it is not possible 
to compare an abstract with a concrete number, or a concrete 
number of one kind with a concrete number of another ; for 
there can exist no measurable relations but between quantities 
of the same kind. 

The series of numbers is indefinite ; but only the first nine 
of them are expressed by different characters, called figures : 
thus, 

Names, one, two, three, four, five, six, seven, eight, nine. 

Figures. 1, 2, 3, 4, 5, 6, 7, 8, 9. 

Besides these, another character is employed, namely 0, 
called the cipher or zero; which has no particular value of 
itself, but by its position is made to change the value of any 
significant figures with which it is connected. 

In the system of numeration now generally adopted, and 
borrowed from the Indians, an infinitude of words and cha- 
racters is avoided, by a simple yet most ingenious expedient, 
which is this : — every figure placed to the left of another 
assumes ten times the value that it would have if it occupied 
the place of the latter. 

Thus, to express the number that is the sum of 9 and 1, or 
ten units, called ten, we place a 1 to the left of a 0, thus 10. 
So again the sum of 10 and 1, or eleven, is represented by 11 ; 
the sum of 11 and 1, or of 10 and 2, called twelve, is repre- 
sented by 12 ; and so on for thirteen, fourteen, fifteen, &c. 
denoted respectively by 13, 14, 15, &c, the figure 1 being all 
along equivalent to ten, because it occupies the second rank. 

In like manner, twenty, twenty-one, twenty-two, &c. are 
represented by 20, 21, 22, because the 2 in the second rank is 
equivalent to twice ten, or twenty. And thus we may proceed 
with respect to the numbers that fall between twenty and three 
tens or thirty 30, four tens or forty 40, five tens or fifty 50, 
six tens or sixty 60, seven tens or seventy 70, eight tens or 
eighty 80, nine tens or ninety 90. After 9 are added to the 90 
(ninety) numbers can no longer be expressed by two figures, 
but require a third rank to the left hand of the second. 

The figure that occupies the third rank, or of hundredths, is 
expressed by the word hundred. Thus 369, is read three 
hundred and sixty-nine ; 428, is read four hundred and twenty- 
eight ; 837, eight hundred and thirty-seven : and so on for all 
numbers that can be represented by three figures. 

But if the number be so large that more than three figures 
are required to express it, then it is customary to divide it into 
periods of three figures each, reckoning from the right hand 
towards the left, and to distinguish each by a peculiar name. 



ARITHMETIC : NOTATION. 3 

The second period is called that of thousands, the third that 
of millions, the fourth that of milliards or billions* the fifth 
that of trillions, and so on ; the terms units, tens, and hundreds, 
being successively applied to the first, second, and third ranks 
of figures from the right towards the left, in each of these 
periods. 

Thus, 1111, is read one thousand one hundred and eleven. 

23456, twenty-three thousands, four hundred and fifty-six. 

421835, four hundred and twenty-one thousands, eight hun- 
dred and thirty-five. 

732846915, seven hundred and thirty-two millions, eight 
hundred and forty-six thousands, nine hundred and fifteen. 

The manner of estimating and expressing numbers we have 
here described is conformable to what is denominated the deci- 
mal notation. But, besides this, there are other kinds invented 
by philosophers, and others indeed in common use ; as the 
duodecimal, in which every superior name contains tivelve 
units of its next inferior name ; and the sexagesimal, in which 
sixty of an inferior name are equivalent to one of its next supe- 
rior. The former of these is employed in the measurement and 
computation of artificer's work ; the latter in the division of a 
circle, and of an hour in time. 

To the head of notation we may also refer the explanation 
of the principal symbols or characters employed to express 
operations or results in computation. Thus, 

The sign -j- (plus) belongs to addition, and indicates that the 
numbers between which it is placed are to be added together. 
Thus, 5 + 7 expresses the sum of 5 and 7, or that 5 and 7 are to 
be added together. 

The sign — (minus) indicates that the number which is 
placed after it is to be subtracted from that which precedes it. 
So, 9 — 3 denotes that 3 is to be taken from 9. 

The sign ^ denotes difference, and is placed between two 
quantities when it is not immediately evident which of them is 
the greater. 

The sign x (into), for multiplication, indicates the product 
of two numbers between which it is placed. Thus 8x5 denotes 
8 times 5, or 40. 

The sign -r- (by), for division, indicates that the number 
which precedes it is to be divided by that which follows it ; 
and the quotient that results from this operation is often 

* It has been customary in England to give the name of billions to millions of 
millions, of trillions to millions of millions of millions, and so on : but the method 
here given of dividing numbers into periods of three figures instead of six, is uni- 
versal on the continent ; and, as it seems more simple and uniform than the other, 
I have adopted it. 



4 ARITHMETIC : NOTATION. 

represented by placing the first number over the second with 
a small bar between them. Thus, 15-J-8 denotes that 15 is to 
be divided by 8, and the quotient is expressed thus y. 

The sign = , two equal and parallel lines placed horizontally, 
is that of equality. Thus, 2 + 3+4=9, means that the sum of 
2, 3, and 4, is equal to 9. 

Inequality is represented by two lines so drawn as to form 
an angle, and placed between two numbers, so that the angular 
point turns towards the least. Thus, 7 > 4, and A > B, indi- 
cate that 7 is greater than 4, and the quantity represented by 
A greater than the quantity represented by B : on the other 
hand, 3 < 5 and C < D indicate that 3 is less than 5, and C 
less than D. 

Colons and double colons are placed between quantities to 
denote their proportionality. So, 3 : 5 : : 9 : 15, signifies that 
3 are to 5 as 9 to 15, or f = T V 

The extraction of roots is indicated by the sign \/, with a 
figure occasionally placed over it to express the degree of the 
root. Thus \/ 4 signifies the square root of 4, <& 27 the cube 
root of 27, \/ 16 the fourth or biquadrate root of 16 ; and so on. 

These characters find their most frequent use in algebra and 
the higher departments of mathematics ; but may, without 
hesitation, be employed whenever they secure brevity without 
a sacrifice of perspicuity. 



Section II. — Addition of Whole Numbers. 

Addition is the rule by which two or more numbers are 
collected into one aggregate or sum. 

Suppose it were required to find the sum of the numbers 
3731, 349, 12487, and 54. It is evident that if we computed 
separately the sums of the units, of the tens, of the hundreds, 
of the thousands, &c. their combined results would still amount 
to the same. We should thus have 15 thousands + 14 hundreds 
+ 20 tens + 21 units, or 15000 + 1400 + 200 + 21; operating 
again upon these, in like manner, rank by rank, we should have 
10 thousands + 6 thousands + 6 hundreds + 2 tens + 1, or 
16621, which is the sum required. 

But the calculation is more commodiously effected by this 

RULE. 

Place the given numbers under each other, so that units 
stand under units, tens under tens, hundreds under hun- 
dreds, &c. 

Add up all the figures in the column of units, and observe 
for every ten in its amount to carry one to the place of tens in 



ARITHMETIC ! SUBTRACTION OF WHOLE NUMBERS. 5 

the second column, putting the overplus figure in the first 
column. 

Proceed in the same manner with the second column, then 
with the third, and so on till all the columns be added up ; the 
figures thus obtained in the several amounts indicate, according 
to the rules of notation, the sum required. 

Note. — Whether the addition be conducted upwards or 
downwards, the result will be the same ; but the operation is 
most frequently conducted by adding upwards. 

Example. — Taking the same numbers as before, 3731 

and disposing them as the rule directs, we have 349 

4 + 74-9-1-1=21, of which we put down the 1 in 12487 
the place of units, and carry the 2 to the tens : 54 

then 2 + 54-8 + 4 + 3=22, of which we put down 

the left hand 2 in the place of tens, and carry the 16621 

other to the hundreds : then 2+4 + 3 + 7= 16, of 

which the 6 is put in the place of hundreds, and the 

1 carried to the thousands. This progress continued will give 

the same sum as before. 

Other Examples. 



57 


6475 


762 


9830 


5389 


2764 


97615 


5937 



103823 



25006 



77756 
3388 
9763 

90257 

181164 



10376786 

789632 

1589 

73 

11168080 



Section III. — Subtraction of Whole Numbers. 

Subtraction is the rule by which one number is taken from 
another, so as to show the difference or excess. 

The number to be subtracted or taken away is called the 
subtrahend ; the number from which it is to be taken, the 
minuend : the quantity resulting, the remainder. 

RULE. 



Write the minuend and the subtrahend in two separate lines, 
units under units, tens under tens, and so on. 

Beginning at the place of units, take each figure in the 
subtrahend from its corresponding figure in the minuend, 
and write the difference under those figures in the same rank 
or place. 

But if the figure in the subtrahend be greater than its cor- 

b 2 



b ARITHMETIC I MULTIPLICATION OF WHOLE NUMBERS. 

responding figure in the minuend, add ten to the latter, and 
then take the figure in the subtrahend from the sum, putting 
down the remainder, as before ; and in this case add 1 to the 
next figure to the left in the subtrahend, to compensate for 
the ten borrowed in the preceding place. 

Thus proceed till all the figures are subtracted. 

Note. — It is customary to place the minuend above the sub- 
trahend ; but this is not absolutely necessary. Indeed, it is 
often convenient in computation to find the difference between 
a number and a greater that naturally stands beneath it ; it is 
therefore, expedient to practise the operation in both ways, so 
that it may, however it occur, be performed without hesi- 
tation. 

Example: . . Minuend 26565874 
Subtrahend 9853642 



Remainder 16712232 



Here the five figures on the right of the subtrahend are each 
less than the corresponding figures in the minuend, and may 
therefore be taken from them, one by one. But the sixth 
figure, viz. 8, cannot be taken from the 5 above it. Yet, as a 
unit in the seventh place is equivalent to 10 in the sixth, this 
unit borrowed (for such is the technical word here employed) 
makes the 5 become 15. Then 8 taken from 15 leaves 7, 
which is put down ; and 1 is added to the 9 in the 7th place 
of the subtrahend, to compensate or balance the 1 which was 
borrowed from the 7th place in the minuend. Recourse 
must be had to a like process whenever a figure in the sub- 
trahend exceeds the corresponding one in the minuend. 

Other Examples. 

From 8217 From 44444 Take 21498 Take 45624 
Take 3456 Take 3456 From 76262 From 80200 



Remains 4761 Remains 40988 Remains 54764 Remains 34576 



Section IV. — Multiplication of Whole Numbers. 

Multiplication of whole numbers is a rule by which we 
find what a given number will amount to when it is repeated 
as many times as are represented by another number.* 

* This definition, though not the most scientific that might be given, is placed 
here, because others depend implicitly, if not explicitly, on proportion, and there- 
fore cannot logically be introduced thus early in the course. 



arithmetic: multiplication of whole NUMBERS. / 

The number to be multiplied, or repeated, is called the multi- 
plicand, and may be either an abstract or a concrete number. 

The number to be multiplied by is called the multiplier, and 
must be an abstract number, because it simply denotes the 
number of times the multiplicand is to be repeated. 

Both multiplicand and multiplier are called factors. 

The number that results from the multiplication is called the 
product. 

Before any operation can be performed in multiplication, the 
learner must commit to memory the following table of products, 
from 2 times 2 to 12 times 12. 



times 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


2 


4 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 


3 


6 


9 


12 


15 


18 


21 


24 


27 


30 


33 


36 


4 


8 


12 


16 


20 


24 


28 


32 


36 


40 


44 


48 


5 


10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


6 


12 


18 


24 


30 


36 


42 


48 


54 


60 


66 


72 


7 


14 


21 


28 


35 


42 


49 


56 


63 


70 


77 


84 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 


88 


96 


9 


18 


27 


36 


45 


54 


63 


72 


81 


90 


99 


108 


10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


110 


120 


11 


22 


33 


44 


55 


66 


77 


88 


99 


110 


121 


132 


12 


24 


36 


48 


60 


12 


84 


96 


108 


120 


132 


144 ( 



It is very advantageous in practice to have this table carried 
on, at least intellectually, to 20 times 20. All the products to 
this extent are easily remembered. 

The learner will perceive that in this table 7 times 5 is equal 
to 5 times 7, or 7 x 5 == 35 = 5 x 7. In like manner that 8 
X 3 = 24 = 3 x 8, 4 x 11 = 44 == 11 X 4, and so of other 
products. This is often made a subject of formal proof, as well 
as that 3x5x8 = 3x8x5 = 5x3x8 = 5x8x3, 
&c. But to attempt the demonstration of things so nearly 
axiomatical as these is quite unnecessary. 



ARITHMETIC I MULTIPLICATION OP WHOLE NUMBERS. 



8 



Previously to exhibiting the rules, let us take a simple 
4827 example, and multiply 4827 by 8. Here placing 
the numbers as in the margin, and multiplying in 
their order 7 units by 8, 2 tens by 8, 8 hundreds by 
8, 4 thousands by 8, the several products are 56 
units, 16 tens, 64 hundreds, 32 thousands : these 
placed in their several ranks, according to the rules 
of notation, and then added up, give for the sum of 
the whole, or for the product of 4827 multiplied by 8 
the number 38616. 



56 
16 
64 
32 

38616 



The same example may be worked thus : 



8 X 


7 = 56^ 




8 X 


20 = 160 




8 X 


800 = 6400 


which is evidently 


8 = 


4000 = 32000 


>the same in effect 
as before. 




38616 



Case I. — To multiply a number, consisting of several figures, 
by a number not exceeding 12. 

Multiply each figure of the multiplicand by the multiplier, 
beginning at the units ; write under each figure the units of the 
product, and carry on the tens to be added as units to the pro- 
duct following. 



Examples. 



Multiply 


4827 


218043 


440052 


8765400 


By 


8 


9 


11 


12 



Products 38616 



1962387 



4840572 



105184800 



Case II. — To perform multiplication when each factor 
exceeds 12. 

Place the factors under each other (usually the smallest at 
bottom), and so that units stand under units, tens under tens, 
and so on. 

Multiply the multiplicand by the figure which stands in the 
unit's place of the multiplier, and dispose the product so that 
its unit's place shall stand under the unit of the multiplicand ; 
then multiply successively by the figure in the place of tens, 
hundreds, &c. of the multiplier, and place the first figure of 
each product under that figure of the multiplier which gave the 
said product. 

The sum of these products will be the product required. 



ARITHMETIC I MULTIPLICATION OP WHOLE NUMBERS. 



Example. 



Multiply 8214356 by 132. 



Multiplicand 8214356 
Multiplier 132 



8214356X2= 
8214356x3 tens = 
8214356X1 hundred = 


• • 
Example 

Prodi 


16428712 
. 24643068 
. 8214356 


8214356X132= . 


. 1084294992 


Other 

Multiply 821436 
by 672576 

4928616 
5750052 
4107180 
1642872 
5750052 
4928616 


S. 

Multiply 8210075 
by 420306 

49260450 
24630225 
16420150 
32840300 


act 3450743782950 


Wuct 552478139136 





Note. — Multiplication may frequently be shortened by sepa- 
rating the multiplier into its component parts or factors, and 
multiplying by them in succession. Thus, since 132 times any 
number are equal to 12 times 11 times that number, the first 
example may be performed in this manner : 



Multiply 8214356^ 
by 11 



And this product 90357916 
by 12 



Product as before 1084294992 



Here one line of 
multiplication, 
and one of addi- 
tion, are saved. 



So, again, the multiplier of the second example, viz. 672576, 
divides into three numbers, 600000, 72000, and 576 ; where, 



10 ARITHMETIC I DIVISION OP WHOLE NUMBERS. 

omitting the cipher, we have 72 = 12 X 6, and 576 = 8 x 72. 
Hence the operation may be performed thus : — 

Multiplicand 821436 

Multiply by 6 in the 6th place 



4928616 
Previous product X 12 . . 59143392 for 72 thousands. 
Second product x 8 .... 473147136 for 576 units. 



Same product as before . . 552478139136 : three lines saved. 
Other modes of contraction will appear as we proceed. 



Section V. — Division of Whole Numbers. 

Division is a rule by which we determine how often one 
number is contained in another. Or, it is a rule by which, 
when we know a product of one of the factors which produced 
it, we can find the other. 

The number to be divided is called the dividend. 
That by which it is divided, the divisor. 
That which results from the division, the quotient, when 
division and multiplication are regarded as reciprocal ope- 
rations. 

The dividend is equivalent to the product. 

The divisor multiplier 

The quotient ■ multiplicand. 

RULE. 

Draw a curved line both on the right and left of the dividend, 
and place the divisor on the left 

Find the number of times the divisor is contained in as many 
of the left hand figures of the dividend as are just necessary, 
and place that number on the right. 

Multiply the divisor by that number, and place the product 
under the above mentioned figures of the dividend. 

Subtract the said product from that part of the dividend under 
which it stands, and bring down the next figure of the dividend 
to the right of the remainder. 

Divide the remainder thus increased, as before; and if at 
any time it be found less than the divisor, put a cipher in the 
quotient, bring down the next figure of the dividend, and con- 
tinue the process till the whole is finished : the quotient figures 
thus arranged will be that required. 



ARITHMETIC I DIVISION OP WHOLE NUMBERS. 11 

Example. 
Divide 743256 by 324. 
Dividend. 
Divisor 324)743256(2294 Quotient 

648 Divisor 324 

Quotient 2294 

1296 
2916 
648 
648 



648 




3045 
2916 




1296 
1296 


P 


Remain .... 




)29754(419^L 

284 


131)135076(1031 T y T 
131 


135 
71 


407 
393 


644 
639 


146 
131 


5 Remain. 


15 Remain. 



Proof 743256 



In these two ex- 
amples the num- 
bers which re- 
main are placed 
over their re- 
spective divisors, 
r*and attached to 
the quotients. — 
The meaning ot 
this will be ex- 
plained when we 
treat of frac~ 
tions. 

Note. — When the divisor does not exceed 12, the operation 
may readily be performed in a single line ; as will appear very 
evident if the following example be compared with the two 
methods of working the first example in multiplication. 

Divide 38616 by 8. 
8)38616(4827 Dividend 38616 

32 Divisor 8 



66 Quotient 4827 
64 

21 

16 Here 8 in 38 go 4 times and 6 over ; these 

carried as 6 tens to the next 6, make 66 : 

56 8 in 66 go 8 times and 2 over ; these car- 

56 ried as 2 tens to the next figure 1 make 21; 

— and so of the rest. 



12 



ARITHMETIC : DIVISION OP WHOLE NUMBERS. 



In division, also, upon the same 
principle as in multiplication, the 
labour may often be abridged by- 
taking component parts of the 
divisor. Thus, in the first example, 
the divisor is equal to 4 times 81, or 
4 times 9 times 9. Hence the di- 
vidend may be divided by 4, 9, and 



9, successively, as in the margin, 
and the result will be the same as 
before. 



Divide 743256 
by 4 



this quotient 185814 
by 9 



and this 
by 



20646 
9 



Quotient 2294 



Since 25 is a fourth part of 100, and 125 the 8th part of 1000, 
it will be easy to multiply or to divide by either of these num- 
bers in a single line — thus, 



To multiply 4827 by 25, put 
two ciphers on the right, 
which is equivalent to multi- 
plying by 100 ; and divide 
by 4. 

4)482700 

120675 Answer. 



To multiply 6218 by 125, put 
3 ciphers, which is equivalent 
to multiplying by 1000 ; then 
divide by 8. 

8)6218000 



777250 Answer. 



To divide 582100 by 25, strike 
off two figures on the right 
hand, which is equivalent to 
dividing by 100 ; then multiply 
by 4. 

5821|00 
4 



23284 



To divide 4567000 by 125, 
strike off three figures on the 
right hand, which is equiva- 
lent to dividing by 1000 ; then 
multiply by 8. 

4567 1000 
8 



36536 



Proof of the first four rules of Arithmetic. 

Simple as these four rules are, it is not unusual to commit 
errors in working them : it is, therefore, useful to possess 
modes of proof. 

1. Now, addition may be proved by adding downwards, as 



2758 

3099 

469 

1029 


2758 
3099 


469 
1029 


7355 




5857 






1498 




7355 



arithmetic: proof of the four rules. 13 

well as upwards, and observing whe- 
ther the two sums agree : or, by 
dividing the numbers, to be added 
into two portions, finding the sum 
of each, and then the sum of those 
two separate amounts. Thus, in the 
margin, the sum of the four num- 
bers is 7355, the sum of the two 
upper ones 5857, of the two lower 
ones 1498, and their sum is 7355, 
the same as before. 



2. The proof of subtraction is effected by adding the re- 
mainder to the subtrahend : if their sum agrees with the 
minuend the work is right, otherwise not. 

3. Multiplication and division reciprocally prove each 
other. 

There is also another proof for multiplication known tech- 
nically by the phrase casting out the nines. Add together 
the numbers from left to right in the multiplicand, dropping 
9 whenever the sum exceeds 9, and carry on the remainder, 
dropping the nines as often as the amount is beyond them ; 
and note the last remainder. Do the same with the multiplier 
and with the product ; then if the product of the first two 
remainders is equal to the last remainder, this is regarded as 
a test that the work is right. Thus, taking the second ex- 
ample in multiplication, the figures in the multiplicand 
amount to 6 above two nines, those in the multiplier to 6 
above three nines, those in the product to above six nines ; 
the product 6 x 6 of the two first excesses is 36, or above 
four nines : the coincidence of the two O's is the proof. It is 
plain, however, that the proof will be precisely the same so 
long as the figures in the product be the same, whatever be 
their order: the proof, therefore, though ingenious, is de- 
fective.* 

A similar proof applies to division. 



* The correctness of this proof, with the exception above specified, may be shown 
algebraically, thus : — put M and JV* = the number of nines in the multiplicand 
and multiplier respectively, m and n their excesses ; then, 9 M + m — the multi- 
plicand, 9 N -f n = the multiplier, and the product of those factors will be = 81 
M N-\- 9 Mn + 9 JYm -f mn ; but the three first terms are each a precise num- 
ber of nines ; because one of the factors in each is so ; these, therefore, being neg- 
lected, there remains m n to be divided by nine ; but m n is the product of the two 
former excesses ; therefore the truth of the method is evident. Q. E. D. 

4 C 



14 ARITHMETIC : WEIGHTS AND MEASURES. 



WEIGHTS AND MEASURES, 

Agreeably to the Act of Uniformity, which took effect 
1st January, 1826. 

The term Measure is the most comprehensive of the two, 
and it is distinguishable into six kinds, viz : — 

r l. Length. 

2. Surface. 

3. Solidity, or Capacity. 
Measures of ^ 4. Force of Gravity, or what is commonly called 

Weight. 

5. Angles. 

6. Time. 

The several denominations of these measures have reference 
to certain standards, which are entirely arbitrary, and conse- 
quently vary among different nations. In this kingdom, 

' Length is a Yard. 
Surface is a Square Yard, the ^^ of an 

Acre. 
C Solidity is a Cubic Yard. 
I Capacity is a Gallon. 
Weight is a Pound. 

The standards of Angular Measure, and of Time, are the 
same in all European, and most other, countries. 

1. Measure of Length. 



The standard of 



12 Inches 


= 1 Foot. 


3 Feet 


= 1 Yard. 


5± Yards 


= 1 Rod or Pole. 


40 Poles 


= 1 Furlong. 



8 Furlongs =1 Mile. 
69 T J T Miles =1 Degree of a Great Circle of the Earth. 

The imperial standard yard, when compared with a pendu- 
lum, vibrating seconds of mean time in the latitude of London, 
in a vacuum, at the level of the sea, is in the proportion of 36 
inches to 39-1393. 

Since the passing of this act, however, some very elaborate 
and scientific experiments of Mr. Francis Baily have shown 
that errors of sufficient moment to be taken into the account 
in an inquiry of this kind render the above proportion inac- 
curate. We do not, in fact, yet know the length of a seconds 



ARITHMETIC : WEIGHTS AND MEASURES. 15 

pendulum at London, vibrating in the circumstances pro- 
posed. 

The following standard yards, made with great accuracy, give 
the annexed results : 

Inches. 

General Lambton's scale, used in India . 35*99934 

Sir Geo. Shuckburgh's scale .... 35*99998 

General Ray's scale 36*00088 

Royal Society's Standard 36*00135 

Ramsden's bar 36*00249 

Its copy, at Marischal College, Aberdeen 36*00244 

An inch is the smallest lineal measure to which a name is 
given ; but subdivisions are used for many purposes. Among 
mechanics the inch is commonly divided into eighths. By the 
officers of the revenue, and by scientific persons, it is divided 
into tenths, hundredths, &c. Formerly it was made to consist 
of 12 parts, called lines, but these have properly fallen into 
disuse. 



Particular Measures of Length. 



A Nail 
Quarter = 
Yard = 
Ell = 
Hand = 
Fathom = 

Link = 



4 
4 
5 
4 
6 



Chain = 100 



Inches 

Nails 

Quarters 

Quarters 

Inches 

Feet 

Inc., 92 
hdths. 
Links 



Used for 
kinds. 



measuring cloth of all 



Used for the height of horses. 

Used in measuring depths. 

Used in Land Measure to facili- 
tate computation of the content, 
10 square chains being equal to 
an acre. 



2. Measure of Surface. 



144 Square Inches 
9 Square Feet 
30j Square Yards 
40 Perches 

4 Roods 
640 Acres 



1 Square Foot. 

1 Square Yard. 

1 Perch, or Rod. 

1 Rood. 

1 Acre. 

1 Square Mile. 



3. Measures of Solidity and Capacity. 



Division I. — Solidity. 

1728 Cubic Inches = 1 Cubic Foot. 
27 Cubic Feet = 1 Cubic Yard. 



16 ARITHMETIC I WEIGHTS AND MEASURES. 



Division II. 

Imperial Measure of Capacity for all liquids, and for all 
dry goods, except such as are comprised in the Third Divi- 
sion : — 

4 Gills = 1 Pint == 34f Cubic Inches, nearly. 

2 Pints = 1 Quart = 69j 

4 Quarts = 1 Gallon = 277\ 
2 Gall. = 1 Peck = 5541 
8 Gall. = 1 Bushel = 2218| 

8 Bush. = 1 Quarter= 10J Cubic Feet, nearly. 

5 Qrs. = 1 Load = 51^ 

The four last denominations are used for dry goods only. 
For liquids several denominations have been heretofore 
adopted, viz. : — For Beer, the Firkin of 9 gallons, the Kilder- 
kin of 18, the Barrel of 36, the Hogshead of 54, and the Butt 
of 108 gallons. These will probably continue to be used in 
practice. For Wine and Spirits, there are, the Anker, 
Runlet, Tierce, Hogshead, Puncheon, Pipe, Butt, and Tun ; 
but these may be considered rather as the names of the casks 
in which such commodities are imported, than as expressing 
any definite number of gallons. It is the practice to guage all 
such vessels, and to charge them according to their actual 
content. 

Flour is sold, nominally, by measure, but actually by weight, 
reckoned at 71b. avoirdupois to a gallon. 

Division III. 

Imperial Measure of Capacity for coals, culm, lime, fish, 
potatoes, fruit, and other goods, commonly sold by heaped 
measure : 

2 Gallons = 1 Peck = 704 ? n , . T , n , 
on n ,-d ui r,o lff ir Cubic Inches, nearly. 
8 Gallons = 1 Bushel =2815^5 

3 Bushels = 1 Sack = 41} n ,. v , i 
12 Sacks = 1 Chald. = 5s| J CublC Fe ^ near1 ^ 

The goods are to be heaped up in the form of a cone, to a 
height above the rim of the measure of at least f of its depth. 
The outside diameter of measures used for heaped goods are to 
be at least double the depth ; consequently not less than the 
following dimensions : — 

Bushel 19| inches. Peck 12^ inches. 

Half-bushel 15j inches. | Gallon 9 J inches 

Half-gallon 7f inches. 



ARITHMETIC : WEIGHTS AND MEASURES. 17 

The Imperial Measures described in the second and third 
divisions were established by act 5 Geo. IV., c. 74. Before 
that time there were four different measures of capacity used in 
England : — 1. For wine, spirits, cider, oils, milk, &c. ; this was 
one-sixth less than the Imperial Measure. 2. For malt liquor; 
this was yV part greater than the Imperial Measure. 3. For 
corn, and all other dry goods not heaped ; this was -^ part less 
than the Imperial Measure. 4. For coals, which did not differ 
sensibly from the Imperial Measure. 

The Imperial Gallon contains exactly lOlbs. avoirdupois of 
pure water, at the temperature of 62° on Fahrenheit's ther- 
mometer, the barometer being at 30 inches ; consequently the 
pint will hold l^lb., and the bushel 80lbs. 

4. Measure or Weight. 

Division I. — Avoirdupois Weight. 

2711 Grains = 1 Dram = 27|J Grains. 

16 Drams = 1 Ounce = 437A — 

16 Ounces = 1 Pound (lb.) = 7000 — 
28 Pounds = 1 Quarter (qr.) 

4 Quarters = 1 Hundred- weight (cwt.) 
20 Cwt. = 1 Ton. 

This weight is used in almost all commercial transactions, 
and in the common dealings of life. 

Particular weights belonging to this division : — 
cwt. qr. lb. 

8 Pounds = 1 Stone Used for Meat. 

14 Pounds = 1 Stone = 14"! 

2 Stone = 1 Tod = 010 

6i Tod = 1 Wey = 1 2 14 

2 Weys = 1 Sack =310 
12 Sacks = 1 Last = 39 



Used in the Wool Trade. 



Division II. — Troy Weight. 

24 Grains = 1 Pennyweight = 24 Grains. 

20 Pennyweights = 1 Ounce = 480 — 

12 Ounces = 1 Pound = 5760 — 

These are the denominations of Troy Weight when used 
for weighing gold, silver, and precious stones (except dia- 
monds). But Troy Weight is also used by apothecaries in 
compounding medicines, and by them the ounce is divided into 
8 drams, and the dram into 3 scruples, so that the latter is equal 
to 20 grains. 

For scientific purposes the grain only is used ; and sets of 

c 2 



18 ARITHMETIC : WEIGHTS AND MEASURES. 

weights are constructed in decimal progression, from 10,000 
grains downwards to y^ of a grain. 

By comparing the number of grains in the Avoirdupois and 
Troy pound and ounce respectively, it appears that the Troy 
pound is less than the Avoirdupois, in the proportion of 14 to 
17 nearly ; but the Troy ounce is greater than the Avoirdupois, 
in the proportion of 79 to 72 nearly. 

The Troy pound is equal to the weight of 22*815 cubic inches 
of distilled water, weighed in air, temperature 62° Fahrenheit, 
barometer at 30 inches. 

oz. dwts. grs. 
1 lb. Avoirdupois = 14 11 15^Troy. 

1 oz. =0 18 5J 

1 dr. =013* 



The carat, used for weighing diamonds, is 3i grains. The 
term, however, when used to express the fineness of gold, has a 
relative meaning only. Every mass of alloyed gold is supposed 
to be divided into 24 equal parts ; thus the standard for coin is 
22 carats fine, that is, it consists of 22 parts of pure gold, and 2 
parts of alloy. What is called the new standard, used for 
watch-cases, &c. is 18 carats fine. 

5. Angular Measure ; or, Divisions of the Circle. 
60 Seconds = 1 Minute. 

60 Minutes = 1 Degree. 

30 Degrees = 1 Sign. 

90 Degrees = 1 Quadrant 

360 Degrees, or 12 Signs = 1 Circumference. 
Formerly the subdivisions were carried on by sixties ; thus, 
the second was divided into 60 thirds, the third into 60 fourths, 
&c. At present the second is more generally divided de- 
cimally into lOths, lOOths, &c. The degree is frequently so 
divided. 

6. Measure of Time. 
60 Seconds = 1 Minute. 

60 Minutes = 1 Hour. 

24 Hours = 1 Day. 

7 Days =1 Week. 

28 Days = 1 Lunar Month. 

28, 29, 30, or 31 Days = 1 Calendar Month. 
12 Calendar Months = 1 Year. 

365 Days = 1 Common Year. 

366 Days = 1 Leap Year. 
365^Days = 1 Julian Year. 
365 D. 5 H. 48 M. 45± S. = 1 Solar Year. 



ARITHMETIC : WEIGHTS AND MEASURES. 



19 



In 400 years, 97 are leap years, and 303 common. 
The same remark, as in the case of Angular Measure, applies 
to the mode of subdividing the second of time. 

Comparison of Measures 

The old ale gallon contained 282 cubic inches. 

The old wine gallon contained 231 cubic inches. 

The old Winchester bushel contained 2150| cubic inches. 

The imperial gallon contains 277*274 cubic inches. 

The corn bushel, eight times the above. 

Hence, with respect to Ale, Wine, and Corn, it will be ex- 
pedient to possess a 

TABLE OP FACTORS, 

For converting old measures into new, and the contrary. 





By Decimals. 


By vulgar Fractions 
nearly. 


Corn 
Measure. 


Wine 
Measure. 


Ale 
Measure. 


Corn 
Mea- 
sure. 


Wine 
Mea- 
sure. 


Ale 
Mea- 
sure. 


To convert old ? . 96g43 1 
measures to new. > | 


1-01704 


3 1 

32 


5 60 

6 5T 


To convert new > 
measures to old 3 


1-20032. I -98324 

I 


32 
3 1 


6 
5 


59 
6TT 



N. B. For reducing the prices, these numbers must all be reversed. 



SIZES OP DRAWING-PAPER 



Wove antique 

Double elephant 

Atlas 

Columbier 

Elephant 

Imperial 

Super-royal 

Royal 

Medium 



4 ft. 

3 ft. 

2 ft. 

2 ft. 

2 ft. 

2 ft. 

2 ft. 

2 ft. 

1 ft. 10 



4 

4 
9 

H 

3* 
5 

3 



X 2 ft. 7 

X 2 ft. 2 

X 2 ft. 2 

X 1 ft. 11 

X 1 ft. ibf 

X 1 ft. 9J 

X 1 ft. 7 

X 1 ft. 7 

X 1 ft. 6 



MISCELLANEOUS INFORMATION. 



1 Aum of hock contains 
1 Barrel, imperial measure 

anchovies 

soap 

herrings 

salmon or eels 



36 gallons 
9981-864 cubic inches 
30 pounds 
256 pounds 
32 gallons 
42 gallons 



20 



ARITHMETIC I WEIGHTS AND MEASURES. 



1 Bushel of coal * . 

flour 
1 Butt of Sherry 
1 Chaldron of coals, with ) 

ingrain 5 

1 Chaldron of coals, without > 

ingrain 5 

1 Chaldron of coals at Newcastle is . 53 cwt. 
[By an act of parliament in 1831, coals within 25 miles of the General Post 
Office, London, must be sold by -weight.] 



88 pounds 

56 pounds 

130 gallons 

104809-572 cubic inches 
99818*64 cubic inches 



1 Clove of wool 


. 






7 pounds 


1 Firkin of butter 




56 pounds 


soap 




64 pounds 


soap 




8 gallons 


1 Fodder of lead, at Stockton 




22 cwt. 


at Newcastle . 




21 cwt. 


at London 




19jcwt. 


1 Gross 




12 dozen 


1 Great gross 








12 gross 


1 Hand 








4 inches 


1 Hogshead of claret . 








58 gallons 


tent 








63 gallons 


1 Hundred of salt 








7 lasts 


1 Keg of sturgeon 








4 or 8 gallons 


1 Last of salt 








18 barrels 


gunpowder 








24 barrels 


beer 








12 barrels 


pot-ash 








12 barrels 


cod-fish 








12 barrels 


herrings 








12 barrels 


meal 








12 barrels 


soap 








12 barrels 


pitch and tar 








12 barrels 


flax 








17 cwt. 


feathers 








17 cwt. 


wool 








4368 pounds 


1 Pack of wool . 








240 pounds 


1 Palm 








3 inches 


1 Pipe of Madeira 








110 gallons 


Cape Madeir 


a 






1 10 gallons 


Teneriffe 








120 gallons 


Bucellas 








140 gallons 


Barcelona 








120 gallons 


Vidonia 








120 gallons 


Mountain 








120 gallons 


Port . 








138 gallons 



ARITHMETIC 



1 Pipe of Lisbon 
1 Pole, Woodland 
Plantation 
Cheshire 
1 Sack of wool 
1 Seam of glass 
1 Span . 
1 Stone of meat 

fish 

(horseman's weight) 

glass 

wool 
1 Tun of vegetable oil 

animal oil 
1 Tod of wool 
1 Wey of cheese, in Suffolk 

in Essex 
1 Wey of wool 



WEIGHTS AND MEASURES. 



21 



140 gallons 
18 feet 
21 feet 
24 feet 
364 pounds 
124 pounds 
9 inches 
8 pounds 
8 pounds 
14 pounds 
5 pounds 
14 pounds 
236 gallons 
252 gallons 
28 pounds 
256 pounds 
336 pounds 
182 pounds 



DIGGING. 



24 Cubic feet of sand, or 18 cubic feet of earth, or 17 cubic 
feet of clay, make 1 ton. 

1 Yard cube of solid gravel or earth contains 18 heaped 
bushels before digging, and 21 heaped bushels when dug. 

27 Heaped bushels make 1 load. 



FRENCH AND ENGLISH WEIGHTS AND 
MEASURES COMPARED. 

The following is a comparative Table of the Weights and 
Measures of England and France, which were published by 
the Royal and Central Society of Agriculture of Paris, in the 
Annuary for 1829, and founded on a Report, made by Mr. 
Mathieu, to the Royal Academy of Sciences of France, on 
the bill passed the 17th of May, 1824, relative to the Weights 
and Measures termed "Imperial," which are now used in 
Great Britain. 



MEASURES OF LENGTH. 



ENGLISH. 

1 Inch (l-36thofayard) 
1 Foot (l-3dof a yard) 
Yard imperial 
Fathom (2 yards) 
Pole, or perch (5 1-2 yards) 
Furlong (220 yards) . 
Mile (1760 yards) 
5 



FRENCH. 

2-539954 centimetres 
3-0479449 decimetres 
0-91438348 metre 
1-82876696 metre 
5-02911 metres 
201-16437 metres 
1609-3149 metres 



22 



ARITHMETIC .* WEIGHTS AND MEASURES. 



FRENCH. ENGLISH. 

1 Millemetre ' 0-03937 inch 

1 Centimetre 0-393708 inch 

1 Decimetre 3-937079 inches 

T39-37079 inches 

1 Metre < 3-2808992 feet 

C. 1-093633 yard 

Myriametre 6-2138 miles 

SQUARE MEASURES. 

ENGLISH. FRENCH. 

1 Yard square 0-836097 metre square 

1 Rod (square perch) .... 25-291939 metres square 

1 Rood (1210 yards square) . 10-116775 are 

1 Acre (4840 yards square) . 0-404671 hectares 



FRENCH. 

1 Metre square 

1 Are . 

1 Hectare . 



ENGLISH. 

1-196033 yard square 
0-098845 rood 
2-473614 acres 



SOLID MEASURES. 



ENGLISH. 

1 Pint (l-8thof a gallon) 
1 Quart (l-4th of a gallon) 
1 Gallon imperial 
1 Peck (2 gallons) 
1 Bushel (8 gallons) 
1 Sack (3 Bushels) 
1 Quarter (8 bushels) 
1 Chaldron (12 Sacks) 



FRENCH. 

0-567932 litre 
1-135864 litre 
4-54345794 litres 
9-0869159 litres 

36-347664 litres 
1-09043 hectolitre 
2-907813 hectolitres 

13-08516 hectolitres 



FRENCH. ENGLISH. 

. T . C 1-760773 pint 

1 L,ltre 1 0-2200967 gallon 

1 Decalitre 2-2009667 gallons 

1 Hectolitre 22-009667 gallons 

WEIGHTS. 

ENGLISH TROT. FRENCH. 

1 Grain (l-24th of a pennyweight) . 0-06477 gramme 

1 Pennyweight (l-20th of an ounce) . 1-55456 gramme 

1 Ounce (l-12th of a pound troy) . 31-0913 grammes 

1 Pound troy imperial .... 0-3730956 kilogramme 



ENGLISH AVOIRDUPOIS. 

1 Drachm (1-1 6th of an ounce) 
1 Ounce (1-1 6th of a pound) 
1 Pound avoirdupois imperial 
1 Hundred-weight (112 pounds) 
1 Ton (20 hundred-weight) . 



FRENCH. 

1*7712 gramme 
28-3384 grammes 

0-4534148 kilogramme 
50-78246 kilogrammes 
1015-649 kilogrammes 



FRENCH. 

1 Gramme . 



1 Kilogramme 



I 



ENGLISH. 

15-438 grains troy 
0-643 pennyweight 
0-03216 ounce troy 
2-68027 pounds troy 
2-20548 pounds avoirdupois. 



ARITHMETIC : FRACTIONS. 



Section VI. — Vulgar Fractions. 

The fractions of which we have already spoken in section 
the 1st, are usually denominated Vulgar Fractions, to distin- 
guish them from another kind, hereafter to be mentioned, called 
decimal fractions. 

A fraction is an expression for a part of a unit, or integer, 
when it represents a whole of any kind. Thus, if a pound 
sterling be the unit, then a shilling will be the twentieth part 
of that unit, and four pence will be four-twelfths of that 
twentieth part. These represented according to the usual 
notation of Vulgar Fractions, will be ■£$ and T 4 ¥ of -i respec- 
tively. ■ 

The lower number of a fraction thus represented (denoting 
the number of parts into which the integer is supposed to be 
divided) is called the denominator ; and the upper figure (which 
indicates the number of those parts expressed by the fraction) 
the numerator. Thus, in the fractions -f? tV? 7 and 15 are 
denominators, 5 and 8 numerators. 

Vulgar fractions are divided into proper, improper, mixed, 
simple, compound, and complex. 

Proper fractions have their numerators less than their de- 
nominators, as f , J-, &c. 

Improper fractions have their numerators equal to, or greater 
than, their denominators, as |, y*> & c - 

Mixed fractions, or numbers, are those compounded of whole 
numbers and fractions, as 7|, 12|, &c. 

Simple fractions are expressions for parts of given units, as 

7? 6"> de- 
compound fractions are expressions for the parts of given 

fractions, as £ of ■§-, f of T 7 T , &c. 

Complex fractions have either one or both terms mixed 

5—12 6 5 
numbers as ^, - ? -i & c . 

Any number which will divide two or more numbers with- 
out remainder is called their common measure. 



Reduction of Vulgar Fractions. 

This consists principally in changing them into a more 
commodious form for the operations of addition, subtraction, 
&c. 

Case 1. — To reduce fractions to their lowest terms : 
Divide the numerator and denominator of a fraction by 



24 ARITHMETIC : FRACTIONS. 

any number that will divide them both, without a remainder ; 
the quotient again, if possible, by any other number : and so 
on, till 1 is the greatest divisor. 

Thus 147 ^- = 114 = - 9 - 8 - = 14 _ 2. where 5 3 7 7 rp- 
spectively are the divisors. 

Or, IU5- = |, by dividing at once by 735. 

Note. — This number 735 is called the greatest common 
measure of the terms of the fraction : it is found thus — Divide 
the greater of the two numbers by the less : the last divisor by 
the last remainder, and so on till nothing remains : the last 
divisor is the greatest common measure required.* 

Case 2. — To reduce an improper fraction to its equivalent 
whole or mixed number. 

Divide the numerator by the denominator, and the quotient 
will be the answer : as is evident from the nature of division. 

Ex. — Let 9 -Jg and ^ *fi De reduced to their equivalent whole 
or mixed numbers. 



43)957(22i| Answer. 274)5480(20 Answer. 

86 548 



97 

86 

11 



* The following theorems are useful for abbreviating Vulgar Fractions : 

Theorems. 

1. If any number terminates on the right hand with a cipher, or a digit divisible 
by 2, the whole is divisible by 2 : for the one which remains in the second place is 
10 ; but 2 measures 10 ; therefore the whole is divisible by 2. 

2. If any number terminates on the right hand with a cipher or 5, the whole is 
divisible by 5 ; for every unit which remains in the second place is 10 ; but 5 mea- 
sures every multiple of 10 ; therefore the whole is divisible by 5. 

3. If the two right hand figures of any number are divisible by 4, the whole is 
divisible by 4 : for every unit which remains in the third place is 100 ; but 4 
measures every multiple of 100 ; therefore the whole is divisible by 4. 

4. If the three right-hand figures of any number are divisible by 8, the whole is 
divisible by 8: for every unit which remains in the fourth place is 1000; but 8 
measures every multiple of 1000 ; therefore the whole is divisible by 8. 

5. If the sum of the digits constituting any number be divisible by 3 or 9, the 
whole is divisible by 3 or 9. 

6. If the sum of the digits constituting any number be divisible by 6, and the 
right-hand digit by 2, the whole is divisible by 6 : for by the data it is divisible both 
by 2 and 3. 

7. If the sum of the 1st, 3d, 5th, &c. digits constituting any number be equal 
to that of the 2d, 4th, 6th, &c. that number is divisible by 1 1 : for if a, b, c, d, e, 



ARITHMETIC '. FRACTIONS. 25 

Case 3. — To reduce a mixed number to its equivalent im- 
proper fraction ; or a whole number to an equivalent fraction 
having any assigned denominator. 

This is, evidently, the reverse of Case 2 ; therefore multiply 
the whole number by the denominator of the fraction, and add 
the numerator (if there be one) to obtain the numerator of the 
fraction required. 

Ex. — Reduce 22J| to an improper fraction, and 20 to a 
fraction whose denominator shall be 274. 

(22 x 43) + 11 = 957 new numerator, and 9 /-j tne first 
fraction. 

20 x 274 = 5480 new numerator, and *■£££ the second 
fraction. 

Case 4. — To reduce a compound fraction to an equivalent 
simple one. 

Multiply all the numerators together for the numerator, and 
all the denominators together for the denominator, of the simple 
fraction requfred. 

If part of the compound fraction be a mixed or a whole num- 
ber, reduce the former to an improper fraction, and make the 
latter a fraction by placing 1 under the numerator. 

When like factors are found in the numerators and denomi- 
nators, cancel them both. 

Ex. — Reduce f of | of % of -J of T 8 T to a simple fraction. 

2X3X5X7X8 2X5X 8 1X5X 8 1X5X 4 20 
3X4X7X9 Xll~~ 4 X 9 Xll"~2 X 9 Xll~~l X 9 XH — 99 

Here the 3 and 7 common to numerator and denominator are 
first cancelled ; then the fraction is divided by 2 ; and then by 
2 again. 

Ex. — Reduce three farthings to the fraction of a pound 
sterling. 

A farthing is the fourth of a penny, a penny the twelfth of a 
shilling, and a shilling the twentieth of a pound. 

Therefore £ of ^ of -^ = ^ = ^ the answer. 

2^ 
Ex. — Simplify the complex fraction -f 

Here, reducing the mixed numbers to improper fractions, we 

have ¥ 3 4 : multiplying by 3, to get quit of the denominator of 

T 

m, n, be the digits, constituting any number, its digits, when multiplied by 11, will 
become 

(8) (7) (6) (5) (4) (3) (2) (1) 

a, a, -{- b, b, + c, c, -f- d, d, -\- e, e, -\- m, m + n n; 

where the odd terms are = to the even. 

D 



26 ARITHMETIC I FRACTIONS. 

the upper fraction, we have tj : multiplying by 5, to get quit 
of the denominator of the lower fraction, we have 4| ; dividing 
both terms of this fraction by 8, there results f for the simple 
fraction required. 

Case 5. — To reduce fractions of different denominators to 
equivalent fractions having a common denominator. 

Multiply each numerator into all the denominators except its 
own, for new numerators ; and all the denominators together 
for a common denominator. 

Ex. — Reduce f, f-, and ■§-, to equivalent fractions having a 
common denominator. 

2 x 7 X 9 = 126} 

6 x 3 X 9 = 162 > the numerators. 
5 x 3 X 7= 105) 

3 x 7 X 9 = 189, the common denominator. 

Hence the fractions are iff, iff, iff, or jf , ff ,*£f , when ab- 
breviated. 

Hence, also, it appears that f exceed f, and that •§- ex- 
ceed f . 

Ex. — Reduce 4 of a penny, and f of a shilling, each to the 
fraction of a pound ; and then reduce the two to fractions having 
a common denominator. 

4 of a penny = 4 of T ^ of ^ = T¥ 4 W = ^ of a pound. 

I of a shilling = f of ^ = j% = ^ = ^% of a pound. 

Hence f of a shilling are 10 times as much as 4 of a penny. 

Note. — Other methods of reduction will occur to the student 
after tolerable practice, and still more after the principles of 
algebra are acquired. 

•Addition and Subtraction of Fractions. 

Rule. — If the fractions have a common denominator, add or 
subtract the numerators, and place the sum or difference as a 
new numerator over the common denominator. 

If the fractions have not a common denominator, they must 
be reduced to that state before the operation is performed. 

In addition of mixed numbers, it is usually best to take the 
sum of the integers, and that of the fractions, separately ; and 
then their sum, for the result required. 



ARITHMETIC : FRACTIONS. 27 



Examples. 
Find the sum of f , -f, and |. 

2 15 1 3 56 I 60 I 61 1J_9 OH 

3"T"7"~T~4 84 T 84T84 84 8 4* 

Take £ of a shilling from --L of a pound sterling. 
I of a shilling = £ of ft = -JL of a pound == ^%. 
Also T V of a pound = fa. Hence ^ — ^ = fa 

AV = n P ence - 

Find the difference between 121 and 8|. 

1Q5 S3 7 7 43 385 2 5 8 127 A 7 

L *6 °T — 6" T — TO 30 - 30 - %0* 



Multiplication and Division of Fractions. 

Rule 1 . — To multiply a fraction by a whole number, mul- 
tiply the numerator by that number, and retain the denomi- 
nator. 

2. To divide a fraction by a whole number, multiply the de- 
nominator by that number, and retain the numerator. 

3. To multiply two or more fractions is the same as to take 
a fraction of a fraction ; and is, therefore, effected by taking 
the product of the numerators for a new numerator, and of 
the denominators for a new denominator. (The product 
is evidently smaller than either factor when each is less than 
unity.) 

4. To divide one fraction by another, invert the divisor, 
and proceed as in multiplication. (The quotient is always 
greater than the dividend when the divisor is less than 
unity.) 

Examples 

1. Multiply J by 2, and divide |- by 5. 

£ X 2 = • = 1 = 11, and f - 5 =-^5* *> ans ' 

2. Multiply 2f by £, and divide | by |. 

Q2 v 3 8 V 3 8 2 nnH 8_!_4 — 8 y 5 10 _ 

*Z X 4 — 3 X 4 —4 — ^? anQ ir~T — ?*4— 7 — 

3. Multiply £2 13s. 4c?. by 31, and divide £4 15s. by 3f 
£2 13s. Ad. =2 +13 + -^ of ft = 2| = 1, and f X 31 

= 1 x J = V = ¥ = H = £9 6s - 8d > 

£4. 1 So _^<?1 — 43_^Ql 19_i_l0 19 v 3 — 57 — 

*,<* lOi. . O-j — <±-% -7- 3 — 4 . -g- — T * To — 40 — 

m =£1 8s. 6d. 



28 ARITHMETIC I FRACTIONS. 

Note. — In the multiplication of Multiply 45| 

mixed numbers, it is often less By 17f 

laborious to perform the multipli- 45 x 7 =315 

cation of each part separately, and 45 x 1 ten = 45. 

collect their sum, as in the margin, I X | = . . . f 

than to reduce the mixed numbers 45 x | = .30 

to improper fractions, and reduce 17 x f = . 12 J 

their product back again to a mixed Product 808^ 

number. 



Section VII. — Decimal Fractions. 

The embarrassment and loss of time occasioned by the com- 
putation of quantities expressed in vulgar or ordinary fractions, 
have inspired the idea of fixing the denominator so as to know 
what it is without actually expressing it. Hence originate two dis- 
positions of numbers, decimal fractions and complex numbers. 
Of the latter, such, for example, as when we express lineal 
measures in yards, in feet (or thirds of a yard), and inches (or 
twelfths of a foot), we shall treat after a few pages. We shall 
now treat of the former. 

Decimal fractions, or substantively, decimals, are fractions 
expressed as whole numbers, but whose values decrease from 
the place of units progressively to the right hand in the same 
decuple or tenfold proportion as the common scale of whole 
numbers increase to the left. They are usually separated from 
the integers by a dot placed between the upper part of the 
figures. Thus, 22 T 7 F expressed according to the decimal notation 
is 22-7. 



Thus, also, -1 is the same as 

•01 

•001 

•0001 



i 

1 

1 
Toro 
1 

10 00 

_J_ 

1 OTTO 
•7 _7_ 

' 10 

_4_3_ 
100 



•43 

•125 T VW 

7-3 %*, 

42-85 42 T «/o 

57-217 57 T y^ 

&c. &c. &c. 



ARITHMETIC : DECIMALS. 29 

The value of a decimal fraction is not altered by ciphers on 
the right hand : for -500, or •£££$, is in value the same as ^, or 
•5, that is -i. 

When decimals terminate after a certain number of figures, 
they are called finite, as -125 = T Wo = h * 958 = ToVo 

137. 

2 5 0* 

When one or more figures in the decimal become repeated, 
it is called a repeating or circulating decimal ; as -333333, &c. 
= i, -66666, &c. m h '428571428571, &c. = |, and many 
others. 

Rules for the management of this latter kind of decimals are 
given by several authors ; but, in general, it is more simple and 
commodious to perform the requisite operations by means of 
the equivalent vulgar fractions, from which circulating decimals 
are educed. 



Reduction of Decimals 

Reduction of Decimals is a rule by which the known parts 
of given integers are converted into equivalent decimals, and 
vice versa. 

Case 1. — To reduce a given vulgar fraction to an equivalent 
decimal. 

Annex ciphers to the numerator, divide by the denominator, 
and the quotient will be the decimal required 



Example, 

1. Reduce J, -J, T 7 j, \\, to equivalent decimals. 
2)1-0 4)3-00 



•5 decimal = | ; -75 decimal = J 



•0000 C 8)11-000 
16^ 64^ 



C 4)7-000 
(4)1.750 



7500 (8) 1-375000 



•4375 decimal = T 7 T ; -171875 decimal = $\. 



d2 



30 ARITHMETIC : DECIMALS. 



2. Reduce -^ and ^± to equivalent decimals. 

•ooooooo 



27 



C 3)4-000000 (7)11 

] 63^ - 

(9)1-333333 (9) 1 



5714285714285 



-148148, &c. -1746031746031, &c. 



decimal = ^ T : decimal = \\. 

These two are evidently circulating decimals, in the former 
of which the figures 148 become indefinitely repeated, in the 
latter the figures 174603. 

3. Reduce 14s. 6d. to the decimal of a pound. 
First 14*. 6d. = }* + i of ft = f £ + ft - » *. 

7*25 
Then f £ = -^r = '725, the decimal required. 

4. Reduce | ■£ to its equivalent decimal 



•o 

399 



57)44-000000(-77192, &c, decimal = |f 



410 
399 

110 
57 

530 
513 

170 
114 

56 

Note. — The above fraction is = 4 • tt> °f which the two 
denominators are both prime numbers (that is, divisible by no 
other number than unity), the entire equivalent decimal is a 
circulator of 18 places, i. e. one less than the last prime .... 
771929824561403508, 7719, &c. over again ad infinitum.* 

* There are many curious properties of fractions A = -14285714, &c. 

whose denominators are prime numbers, one of 2 __ «28571428 <fcc. 

which may be here shown in reference to frac- 3 '42857142 &LC 

tions having the denominator 7. The circulating J " " . __„' « 

figures of the equivalent decimals are precisely the T == *&714Zoo7, CLC 

for i, 2, &c. and in the same order: the cir- 4 = '71428571, &c. 



same, 



T> 



*• 



culate merely commences at a different place for 7 = -85714285, &c. 

each numerator. 



ARITHMETIC : DECIMALS. 



31 



Case 2. — Any decimal being given to find its equivalent 
vulgar fraction ; or to express its value by integers of lower de- 
nominations. 

When the equivalent vulgar fraction is required, place under 
the decimal as a denominator a unit with as many ciphers as 
there are figures in the proposed decimal ; and let the fraction 
so constituted be reduced to its lowest terms. 

Or, if the value of the decimal be required in lower deno- 
minations, multiply the given decimal by the value of its in- 
teger in the next inferior order ; and point off, from right to 
left, as many figures of the product as there were places in the 
given decimal. 

Multiply the decimal last pointed off by the value of its 
integer, in the next inferior order, pointing off the same number 
of decimals as before : and thus continue the process to the 
lowest integer, or until the decimals cut off become all ciphers ; 
then will the several numbers on the left of the separating 
points, together with the remaining decimal, if any, express the 
required value of the given decimal. 

Examples 



Find the vulgar fractions equivalent to '25 and -375. 

answers. 



.25 = Afr = i; and -375 = ^ 



3 
■8' 



2. Find the value in shillings, &c. of '528125 of a £. 

.528125 
20 



10-562500 
12 



6-7500=6| 



>Ans. 10s. 6fd. 



3. Find the value of -74375 of an acre. 
•743751 
4 



2-97500 
40 



39-000 



JLns. 2 roods 39 perches. 



32 



ARITHMETIC I DECIMALS. 



Addition and Subtraction of Decimals. 

These are performed precisely as in whole numbers, the 
numbers being so arranged that units stand under units, tens 
under tens, &c. or, which amounts to the same thing, that the 
decimal points stand under one another. Thus, 



Add 
together 




From 2486-173 
Take 14*56789 



Remains 2471-60511 



Sum 465-5692 



Proof 2486-17300 



Multiplication and Division of Decimals. 

Here, again, the operations are performed as in integers. 
Then, in multiplication, let the product contain as many decimal 
places as there are in both multiplier and multiplicand, ciphers 
being prefixed, if necessary, to make that number ; and, in 
division, point off as many decimals in the quotient as the num- 
ber in the dividend (including the ciphers supplied, if there be 
any) exceeds that in the divisor 



Examples. 



1. Multiply 43-7 by 3-91, and 2'4542 by -0053. 



43-71 
39-1 



437 
3933 
1311 

170-867 



Here 4 3*7x3-91 



4_3_7 
10 



V 3 91 



U70867 — 170 867 

? Tooo — uu Tooo' 



as in the 
operation. 



decimal 



2-4542 
•0053 

73626 
122710 

•01300726 



Here one cipher 
is prefixed to 
make the requi- 
site number of 
decimals in the 
product 



ARITHMETIC I DECIMALS. 



33 



2. Divide 172-8 by -144, and 192 by 5'423. 
•144)172-S(1200- quotient 5-423)192'000(35'40475 

144 16269 



288 
288 



.00 



In the first of these examples the 
two ciphers brought down, together 
with the decimal 8, make the num- 
ber of decimals in the dividend the 
same as in the divisor, therefore the 
quotient is entirely of integers. In 
the second example, 3, the decimal 
places in the divisor, taken from 8, 
the decimal places in the dividend 
(including those brought down), 
leave 5 for the decimal places in the 
quotient. 



29310 
27115 

21950 
21692 



25800 
21692 



41080 
37961 

31190 
27115 

4075 



Section VIII. — Complex Fractions used in the Jirts and 
Commerce. 

In the arts and in commerce, it is customary to assume a 
series of units having a constant relation to each other, so that 
the units of one denomination become fractions of another. 
One farthing, for example, is | of a penny, 1 penny T \ of a 
shilling, 1 shilling ^ of a pound, or i °f a guinea. One 
lineal inch, again, is r V of a foot, 1 foot i of a yard, and so 
on, according to the relations expressed in the tables at the 
end of the fifth section. The arithmetical operations on 
complex numbers of these kinds are usually effected by 
simpler rules than those which apply to vulgar fractions 
generally ; of which it will therefore be proper here to specify 
a few. 



Reduction. 
Here we have two general cases : 

Case 1. — When the numbers are to be reduced from a higher 
denomination to a lower. 



34 ARITHMETIC : COMPLEX NUMBERS. 

1. Multiply the number in the higher denomination by as 
many of the next lower as make an integer, or one, in that 
higher, and set down the product. 

2. To this product add the number, if any, which was in this 
lower denomination before ; and multiply the sum by as many 
of the next lower denomination as make an integer in the pre- 
sent one. 

3. Proceed in the same manner through all the denominations 
to the lowest, and the number last found will be the value of 
all the numbers which were in the higher denominations taken 
together. 

Case 2. — When the numbers are to be reduced from a lower 
denomination to a higher. 

1. Divide the given number by as many of that denomi- 
nation as make one of the next higher, and set down what 
remains. 

2. Divide the quotient by as many of this as make one of the 
next higher denomination, and set down what remains in like 
manner as before. 

3. Proceed in the same manner through all the denominations 
to the highest ; and the quotient last found, together with the 
several remainders, if any, will be of the same value as the first 
number proposed. 

The method of proof is to work the question back again. 



Examples. 

1. Reduce £14 to shillings, pence and farthings; and 24316 
farthings into pounds, &c. 



14 
20 


24316 

-r-4 


280 shillings 
12 


6079 pence 

-r-12 


3360 pence 
4 


506 7 
H-20 


13440 farthings 


£25 6s. 7d. 



ARITHMETIC I COMPLEX NUMBERS. 



35 



2. Reduce 22 Ac. 3 R. 24 P. into perches : and 52187 
perches into acres. 

22 3 24 52187 

4 -h40 



91 roods 
40 

3664 perches 



1304 27 
-J-4 



Ac. 326 R. 27 P. 



•Addition and Subtraction. 

Rule. — Write, one under the other, the parts which have 
the same denomination, and operate successively on each of 
them, beginning with the smallest. If any sum surpass the 
number of units necessary to form one or more units of the 
next superior order, put down the excess, and carry on the 
other. Proceed similarly with regard to what is borrowed in 
subtraction. 



Examples. 



Add 



£ s. 

368 10 

257 10 

88 11 

33 10 

12 13 

8 8 



d. 
3 
5 

4* 


5 



ib. 

Add 14 
17 
15 

2 

13 

4 



oz. divt. gr. 

6 12 13 
5 3 12 

9 16 

7 15 20 
2 10 19 

1 5 21 



lb. oz. d-wt. gr. 

Add 10 8 11 17 

42 5 16 12 

12 2 14 18 

51 6 22 

24 9 17 17 

29 4 18 22 



Sum 769 4 2 Sum 66 11 18 5 Sum 171 2 12 



£ s. 


d. 


£ 


s. d. 


lb. oz. divt. gr. 


From 16 12 


H 


From 21 


13 4| 


From 18 9 10 8 


Take 10 11 


H 


Take 18 


9 8| 


Take 9 10 15 20 



Rem. 6 1 2| 



Rem. 3 3 8j 



Rem. 8 10 14 12 



36 ARITHMETIC ! COMPLEX NUMBERS. 

Multiplication and Division. 

1. In Compound Multiplication, place the multiplier under 
the lowest denomination of the multiplicand. — Multiply the 
number in the lowest denomination by the multiplier, and find 
how many integers of the next higher denomination are con- 
tained in the product, and write down what remains. — Carry 
the integers, thus found, to the product of the next higher de- 
nomination, with which proceed as before ; and so on, through 
all the denominations to the highest ; and this product, together 
with the several remainders, taken as one number, will be the 
whole amount required. 

If the multiplier exceed 12, multiply successively by its 
component parts ; as in the following examples. 

2. In Compound Division, place the divisor and dividend 
as in simple division. — Begin at the left hand or highest de- 
nomination of the dividend, which divide by the divisor, and 
write down the quotient. — If there be any remainder after this 
division, find how many integers of the next lower denomi- 
nation it is equal to, and add them to the number, if any, 
which stands in that denomination. — Divide this number so 
found, by the divisor, and write the quotient under its proper 
denomination. — Proceed in the same manner through all the 
denominations to the lowest, and the whole quotient, thus found, 
will be the answer required. 

Examples. 

£ s. d. a. r. p. 

1. Multiply 4 17 6j by 441, and 3 2 14 by 531. 



£ s. d. 
4 17 6£ 
9 x 7 X 7 = 441 9 


Ans. 


10 


35 3 20 
10 


43 17 10J 

7 


358 3 for 100 
5 


307 4 llf 

7 


1793 3 for 500 
107 2 20 3 times 10 
3 2 14 1 top line. 


Jlns. £2150 14 10j 




1904 3 34 



ARITHMETIC I COMPLEX NUMBERS. 37 



£ S. d. 
2 Divide 521 18 6 by 432 

432 = 12 X 12 X 3. 

Therefore, by short division 



£521 18 6 
— 12 


-*. 


43 9 10J 
•12 


3 12 5f x J a farthing. 

H-3 


Quotient £ 1 


4 If x f- of a farthing 




By long division : 


£ s. 
432)521 18 
432 


d. £ s. d. 

6(1 4 If x 4 of a farthing. 


89 
20 




432)1798(4 
1728 


360 30 5 

TTJ— 3-6— T 


70 
12 




432)846(1 
432 




414 
4 




432)1656(3 
1296 




360 




7 


E 



38 arithmetic: duodecimals. 



Duodecimals. 



Fractions whose denominators are 12, 144, 1728, &c. are 
called duodecimals ; and the division and sub-division of the 
integer are understood without being expressed, as in decimals. 
The method of operating by this class of fractions is principally 
in use among artificers, in computing the contents of work, of 
which the dimensions were taken in feet, inches, and twelfths of 
an inch. 

Rule. — Set down the two dimensions to be multiplied to- 
gether, one under the other, so that feet shall stand under feet, 
inches under inches, &c. Multiply each term in the multipli- 
cand, beginning at the lowest, by the feet in the multiplier, and 
set the result of each immediately under its corresponding term, 
observing to carry 1 for every 12, from the inches to the feet. 
In like manner, multiply all the multiplicand by the inches of 
the multiplier, and then by the twelfth parts, setting the result 
of each term one place removed to the right hand when the 
multiplier is inches, and two places when the parts become the 
multiplier. The sum of these partial products will be the an- 
swer required. 

Or, instead of multiplying by the inches, &c. take such parts 
of the multiplicand as these are of a foot. 



Examples. 

1. Multiply 4/ 7 inc. into 8/ 4 inc. 

4/ 7 i. or, 4/ 7 i. 

8 4 8 



36 
1 


8 
6 


4 


38 


2 


4 



36 8 
4 = | 1 6J 



38 2J 



Here the 2 which stands in the second place does not denote 
square inches, but rectangles of an inch broad and a foot long, 
which are to be added to the square inches in the third place, so 
that (2 x 12) X 4 = 28 are the square inches, and the product 
is 38 square feet, 28 square inches 



ARITHMETIC : DUODECIMALS. 

2. Multiply 35/ 4£ inc. into 12/ 3 J inc. 

35 4 6 or, 35 4^ 

12 3 4 12 



424 6 

8 10 1 

11 9 


6 





434 3 11 









424 6 
3=J 8 10 lj 
£=4 of 3 11 9J 



434 3 11 



Here, again, the product is 435 square feet, + (3 x 12) + 
1 1 inches, or 434 square feet, 47 square inches. And this man- 
ner of estimating the inches must be observed in all cases 
where two dimensions in feet and inches are thus multiplied 
together 



Section IX.— Powers and Roots. 



A power is a quantity produced by multiplying any given 
number, called the root or radix, a certain number of times con- 
tinually by itself. The operation of thus raising powers is 
called involution. 

3 = 3 is the root, or 1st power of 3; 
3 x 3 = 3 2 =9, is the 2d power, or square of 3 
3 X 3 x 3 = 3 3 =27, is the 3d power, or cube of 3 
3 x 3 X 3 x 3= 3 4 =81, do. 4th power or biquadrate of 3 
&c. &c. &c. 

Table of the first Nine Powers of the first Nine Numbers. 



2d 3d 



16 



27 



64 



25 125 



36 



49 



64 



81 



216 



343 



512 



729 



4th 5th I 6th 



16 



32 



81 



256 



625 



1296 



2401 



4096 



6561 



243 



1024 



3125 



7776 



64 



729 



4096 



15625 



46656 



16807 117649 



32768 262144 



59049 531441 



7th 



128 



8th 



256 



9th 



512 



2187 



16384 



78125 



279936 



823543 



6561 



65536 



390625 



1679616 



5764801 



2097152 16777216 134217728 



19683 



262144 



1953125 



10077696 



40353607 



4782969 



43046721387420489 



40 arithmetic: square root. 

So again, fxf=|=square off; |xf =2 8 T =cube of f ; ^ T X 
| = .j.6, biquadrate off ; and so of others. Where it is evident, 
that while the powers of integers become successively larger 
and larger, the powers of pure or proper fractions become suc- 
cessively smaller and smaller. 



Evolution. 



Evolution, or the extraction of roots, is the reverse of in- 
volution. 

Any power of a given number may be found exactly ; but 
we cannot, conversely, find every root of a given number 
exactly. Thus, we know the square root of 4 exactly, being 
2 ; but we cannot assign exactly the cube root of 4. So again, 
though we know the cube root of 8, viz. 2, we cannot exactly 
assign the square root of 8. But of 64 we can assign both the 
square root and the cube root, the former being 8, the latter 4. 

By means of decimals we can in all cases approximate the 
root to any proposed degree of exactness. 

Those roots which only approximate are called surd roots, or 
surds, or irrational numbers ; as s/2, ^5, V 9j &c., while 
those which can be found exactly are called rational ; as \/9 
= 3, ^125=5, V 16 = 2. 



To extract the square root. 

Rule. — Divide the given number into periods of two figures 
each, by setting a point over the place of units, another over 
the place of hundreds, and so on over every second figure, both 
to the left hand in integers, and to the right hand in decimals. 

Find the greatest square in the first period on the left hand, 
and set its root on the right hand of the given number, after 
the manner of a quotient figure in Division. 

Subtract the square thus found from the said period, and to 
the remainder annex the two figures of the next following 
period, for a dividend. 

Double the root above mentioned for a divisor, and find how 
often it is contained in the said dividend, exclusive of its right- 
hand figure ; and set that quotient figure both in the quotient 
and divisor. 

Multiply the whole augmented divisor by this last quotient 



ARITHMETIC I SQUARE ROOT. 41 

figure, and subtract the product from the said dividend, 
bringing down to the next period of the given number for a 
new dividend. 

Repeat the same process, viz. find another new divisor, by 
doubling all the figures now found in the root ; from which, 
and the last dividend, find the next figure of the root as 
before ; and so on through all the periods, to the last.* 

Note. — The best way of doubling the root, to form the 
new divisors, is by adding the last figure always to the last 
divisor, as appears in the following Examples. — Also, after 
the figures belonging to the given number are all exhausted, 
the operation may be continued into decimals at pleasure, 
by adding any number of periods of ciphers, two in each 
period. 



Examples. 
1. Find the square root of 17*3056. 



17*3056(4-16 the root : in which the 
16 number of decimal places is 

the same as the number of 

130 decimal periods into which 

81 the given number was di- 

vided. 



81 

1 



826 



4956 
4956 



* The reason for separating the figures of the dividend into periods or portions 
of two places each, is, that the square of any single figure never consists of more 
than two places ; the square of a number of two figures of not more than four 
places, and so on. So that there will be as many figures in the root as the given 
number contains periods so divided or parted off. 

And the reason of the several steps in the operation appears from the algebraic 
form of the square of any number of terms, whether two or three, or more. Thus, 
352 = 30 2 + 2 . 30 . 5 + 5 2 or generally (a -f bf = a? + 2 a b + b 2 = cfl + 
(2 a -|- b) b, the square of two terms ; where it appears that a is the first term of 
the root, and b the second term ; also a the first divisor, and the new divisor is 2 a 
+ b, or double the first term increased by the second. And hence the manner of 
extraction is as in the rule. 

E 2 



42 



ARITHMETIC : EVOLUTION. 



2. Find the square root of 2, to six decimals. 
2(1-414213 root. 



24 100 
4 96 


281 
1 


400 

281 


2824 
4 


11900 
11296 


28282 
2 


60400 
56564 


28284 


1 383600 
1 282841 



2828423 



10075900 
8485269 

1590631 



3. Find the square root of T 5 g. 
T % = -416666666, &c. 

0-416666(0-64549, &c. 
36 



124 
4 



566 
496 



1285 
5 



7066 
6425 



12904 
4 



64166 
51616 



129089 



1255066 
1161801 



93265 



ARITHMETIC I EVOLUTION. 43 

Note. — In cases where the square roots of all the integers 
up to 1000 are tabulated, such an example as the above may- 
be done more easily by a little reduction. Thus %/ -^ = 

7-7459667 
*0A • if) = •* = A ^ 6 ° =—17- ='645497, &c. 



Cube and higher roots. 

The rules usually given in books of arithmetic for the cube 
and higher roots, are very tedious in practice : on which 
account it is advisable to work either by means of approxi- 
mating rules, or by means of logarithms. The latter is, gene- 
rally speaking, the best method. We shall merely present 
here Dr. Hutton's approximating rule for the cube root, which 
may sometimes be serviceable when logarithmic tables are not 
at hand. 

Rule. — By trials take the nearest rational cube to the given 
number, whether it be greater or less, and call it the assumed 
cube. 

Then say, by the Rule of Three, as the sum of the given 
number and double the assumed cube, is to the sum of the 
assumed cube and double the given number, so is the root of the 
assumed cube, to the root required, nearly. Or, as the first sum 
is to the difference of the given and assumed cube, so is the as- 
sumed root, to the difference of the roots, nearly. 

Again, by using, in like manner, the cube of the root last 
found as a new assumed cube, another root will be obtained 
still nearer. And so on as far as we please ; using always the 
cube of the last found root, for the assumed cube. 



Example. 

To find the cube root of 21035-8. 

Here we soon find that the root lies between 20 and 30, and 
then between 27 and 28. Taking therefore 27, its cube is 
19683, which is the assumed cube. Then 



44 ARITHMETIC I CUBE ROOT, &C. 

19683 21035-8 

2 2 



39366 
21035-8 


42071.6 
19683 


60401-8 


: 61754-6 : 

27 




4322822 
1235092 



27 : 27-6047 



60401-8)1667374-2(27-6047 the root nearly. 
459338 Again, assuming 27*6 

36525 and working as before, 
284 the root will be found 
42 27-60491. 



Section X. — Proportion. 

Two magnitudes may be compared under two different 
points of view, that is to say, either by inquiring what is the 
excess of one above the other, or how often one is contained 
in the other. The result of this comparison is obtained by 
subtraction in the first case, by division in the second. The 
ratio of two numbers is indicated by the quotient resulting 
from dividing one by the other. Thus 3 may be regarded as 
the ratio of 12 to 4, since ^ or 3 is the quotient of the numbers 
12 and 4. 

The first of two numbers constituting a ratio is called the 
antecedent, the second the consequent. 

The difference of two numbers is not changed by adding 
one and the same number to each, or by subtracting the same 
number from each. 

Thus 12 —5 = (12 + 2 — (5 + 2)= 14 — 7 = (12 — 2) 
— (5-^2) = 10 — 3. 

In like manner, a ratio is not changed by either multiplying 
both its terms, or dividing both its terms by the same 
number. 

Thus y = (V . f =) ft - (¥ + * -) fr 

Since surds enter arithmetical calculations by means of 



ARITHMETIC : PROPORTION. 45 

their approximate values, a sufficiently precise idea may be ob- 

/3 1*73205 
tained of their ratio : thus,-— =-—-—. This ratio is often 

y/2 1*41421 

\/12 
commensurable even with respect to surds : as —-—= 

^/ 6 

^(12 — 3) y4_2 

Equality of differences, or equidifference, is a term used to 
indicate that the difference between two numbers is the same as 
the difference between other two, or other two. Such, for 
example, as 12 — 9 = 8 — 5 = 7 — 4. 

Equality of ratios, or proportion, is similarly employed to 
denote that the ratio of two numbers is the same as that 
between two others. Thus 20 and 10, 14 and 7, have 2 
for the measure of the ratio : we have therefore a proportion 
between 20 and 10, 14 and 7, which is thus expressed, 20 : 10 
: : 14 : 7; and thus read 20 are to 10 as 14 are to 7. The same 
proportion may also be represented thus, y& = y. Though, 
by whatever notation it be represented, it is best to read or 
enumerate it as above. It is true, however, that in all cases 
when two fractions are equal, the numerator of one of them is to 
its denominator, as the numerator of the other to its denominator. 

In a proportion, as 20 : 10 : : 14 : 7, the second and third 
terms are called the means, the first and fourth the extremes. 

When the two means are equal, the proportion is said to be 
continued. Thus 3 : 6 : : 6 : 12 are in continued proportion. 
This is usually expressed thus -f* 3 : 6 : 12 ; and the second 
term is called the mean proportional. 

In the case of equidifference, as 12 — 9 = 7 — 4, the sum of 
the extremes (12 and 4) is equal to that of the means (9 and 7). 
In like manner in a proportion, as 20 : 10 : : 14 : 7 , the product 
of the extremes (20 and 7) is equal to that of the means (10 and 
14). The converse of this likewise obtains, that if 20 x 7 = 
10 x 14, then 20 : 10 : : 14 : 7. 

Hence, 1. If there be four numbers, 5, 3, 15, 9, such that the 
products 5x9 and 3x15 are found equal, we may infer the 
equality of their ratios, or the proportion | = y, or 5 : 3 : : 
15:9. So that a proportion may always be constituted with 
the factors of two equal products. 

2. If the means are equal, their product becomes a square ; 
therefore the mean proportional between two numbers is equal 
to the square root of their product. Thus, between 4 and 9 the 
mean proportional is </(4 x 9) = 6. 

3. If a proportion contain an unknown term, such, for 
example, as 5 : 3 : : 15 : the unknown quantity ; since 5 times 

8 



46 ARITHMETIC I PROPORTION. 

the unknown quantity must be equal to 3 x 15 or 45, that 
quantity itself is equal to 45 ~ 5 or 9. Generally one of the 
extremes is equal to the product of the means divided by the 
other extreme ; and one of the means is equal to the product of 
the extremes divided by the other mean. 

4. We may, without affecting the correctness of a proportion, 
subject the several terms which compose it to all the changes 
which can be made, while the product of the extremes remains 
equal to that of the means. Thus, for 5 : 3 : : 15 : 9, which 
gives 5 x 9 = 3 x 15, we may 

I. Change the places of the means without changing those 
of the extremes, or change the places of the extremes with- 
out changing those of the means : this is denoted by the term 
alternando. 



Thus 


5:3: 


: 15 : 9 


become 


j 5 : 15 : 


: 3 : 9 


or 


9:3: 


: 15 : 5 


or 


9 : 15 : 


: 3 : 5 



II. Put the extremes in the places of the means ; this is 
called invertendo. 

3 : 5 : : 9 : 15 

III. Multiply or divide the two antecedents or the two con- 
sequents by the same number. 

It also appears, with regard to proportions, that the sum or 
the difference of the antecedents is to that of the consequents, as 
either antecedent is to its consequent. 

And, that the sum of the antecedents is to their difference, as 
the sum of the consequents is to their difference. 

^ u 5 t 15 _ 5 _ 15 and 5 + 15 _ 5^15 
Thus -fy ~ 3 - t > and 3+^-—' 

If there be a series of equal ratios represented by f = y* = 
V = i| we shall have t + 1 ^^ - « - 1 - ■ V - **■ 

Therefore, in a series of equal ratios, the sum of the antecedents 
is to the sum of the consequents, as any one antecedent is to its 
consequent. 

If there be two proportions, as 30 : 15 : : 6 : 3, and 2 : 3 : : 4 : 6, 
then multiplying them term by term we shall have 30 x 2 : 
15x3::6x4:3x6, which is evidently a proportion, be- 
cause 30X2X3X6=15X3X6X4= 1080. 

Thus, also, any powers of quantities in proportion are in 
proportion ; and conversely of the roots. 
2:3: 
2:3: 



2 : 3 




2 2 : 3 2 : : 6 2 : 9 2 

2 3 : 3 3 : : 6 3 : 9 3 



ARITHMETIC '. RULE OF THREE. 47 



Rule of Three. 

When the elements of a problem will form a proportion ot 
which the unknown quantity is the last term, a simple calcula- 
tion will determine it, and the problem is said to belong to the 
Golden Rule, or Rule of Three. The operation is regulated 
by the foregoing principles of proportion. 

Of the three given numbers, two are called the terms of 
supposition, and the other the term of demand. Now if the 
term of demand be greater or less than the other term of the 
same kind, and the question require the term sought to be re- 
spectively greater or less than the other, the question belongs to 
the Rule of Three direct : otherwise it belongs to the Rule of 
Three inverse. 

For the Rule of Three direct we have this 

Rule. — Write the three given terms in the following order, 
viz. let that which implies or asks the demand be put in the 
third place, and the other of the same kind in the first : then 
will the remaining term, which is similar to the fourth or re- 
quired one, occupy the second place. Having thus disposed 
the numbers, called stating the question, reduce the first and 
third terms to one and the same denomination ; and if the 
second term be a compound one, reduce it to the lowest name 
mentioned. Multiply the second and third terms together, 
and divide the product by the first, and the quotient will be 
the answer, in the same denomination to which you reduced the 
second term. 

When the second term is a compound one, and the third a 
composite number, it is generally better to multiply the second 
term, without any previous reduction, by the component parts 
of the third, as in compound multiplication, after which divide 
the compound product by the first term, or, by its factors. (Here 
the first and third terms are homogeneous, in a given ratio, the 
second and fourth in the same.) 



For the Inverse Rule. 

State the question and reduce the terms in the direct rule : 
then multiply the first and second terms together and 
divide the product by the third, and the quotient will be the 
answer. 



48 



ARITHMETIC : RULE OP THREE. 



Examples in the Direct Rule. 



1. If 3 gallons of 


brandy 


2. How much brandy may 


cost 19*., what will 126 gal- 


be bought for 39/. 18s. ; at 


lons cost at the same rate ? 


the rate of 3 gallons for 19 






shillings ? 


gal, s. gal. 




s. gal. £ s. 


3:19 :: 126 : ? 




19 : 3: : 39 18 : ? 


19 




20 


1134 




798,9. 


126 




3 


2|0 




gal. 


3)2394(7918 




19)2394(126 Ans. 


21 




16 


39/. 18*. 


*flns. 





29 




49 


27 




38 


24 




114 


24 




144 



3. If 21 yards of cloth cost 
24/. 105. what will 160 yards 
cost? 

£ s. 

Here, 21 : 24 10 : : 160 :? 
X 4 

4X4X10 = 160 

98 
X4 



392 
X 10 

3920 
•4-3 

1306 13 4 

+7 



£186 13*. Ad, Jins. 



4. If selling by cloth at 
1/. 2s. per yard, 10 per cent, 
is gained, what would be gain- 
ed if it had been sold at 1/. 5s. 
per yard ? 

£ s. £ s. 

Here, 1 2:110::1 5:? 
20 20 



22 



25 
XllO 

2)2750 



22 



11)1375 



Amount £125 
Deduct 100 



Gain per cent. £25 



ARITHMETIC ! RULE OP THREE. 



49 



5. What is the simple inte- 
rest of 560/. for 5 years, at 4 
per cent, per annum ? 
£ £ £ 
Here, 100 : 4 : : 560 
4 

100 ) 2240 



£22-4 
20 



80 

Interest for one year £22 8s. 
y. £ s. y. 
Then 1 : 22 8 : : 5 
5 



£112 Answer. 



6. If 100 workmen can 
finish a piece of work in 12 
days, how many men work- 
ing equally hard would have 
finished it in 3 days ? 

This example is manifestly 
in the Rule of Three inverse. 

d. vf. h. 

Hence, 12 : 100 : : 3 : ? 
12 

3 ) 1200 

Answer 400 workmen. 



A distinct rule is usually given for the working of problems 
in Compound Proportion ; but they may generally be solved 
with greater mental facility by means of separate statings. 
Thus: 

8. If a family of 9 persons 
spend £480 in 8 months, how 
much will serve a family (liv- 
ing upon the same scale) of 24 
persons 1 6 months ? 



7. If a person travel 300 
miles in 10 days of 12 hours 
each, in how many days of 16 
hours each may he travel 600 
miles ? 

First, if the days were of 
the same length, it would be, 
by direct proportion, 

m. d. m. 

As 300 : 10 : : 600 : 20 days. 

But these would be days of 
12 hours each, instead of 16, 
of which fewer will be re- 
quired. Hence, by inverse 
proportion, 

As 12: 20:: 16: l -^= 15. 



16 



p. £ 
First, as 9 : 480 : 

=£1280. 



24 



480X24 



But this would only be the 
expense for 8 months. Hence, 
again, 

m. £ m. £ 

As 8 :1280: : 16 : 2560, the 
expense of the 24 persons for 
16 months. 
So that the answer is 15 days. 

Note. — The Rule of Three receives its application in ques- 
tions of Interest, Discount, Fellowship, Barter, &c. 

F 



50 PROPERTIES OP NUMBERS. 



Properties of Numbers. 

To render these intelligible to the student, we shall here col- 
lect a few definitions. 

1. An unit, or unity, is the representation of any thing con- 
sidered individually, without regard to the parts of which it is 
composed. 

2. An integer is either a unit or an assemblage of units ; 
and a fraction is any part or parts of a unit. 

3. A multiple of any number is that which contains it some 
exact number of times. 

4. One number is said to measure another, when it divides 
it without leaving any remainder. 

5. And if a number exactly divides two, or more numbers, it is 
then called their common measure. 

6. An even number is that which can be halved, or divided 
into two equal parts. 

7. An odd number is that which cannot be halved, or which 
differs from an even number by unity. 

8. A prime number is that which can only be measured by 
l,or unity. 

9. One number is said to be prime to another when unity is 
the only number by which they can both be measured. 

10. A composite number is that which can be measured by 
some number greater than unity. 

11. A perfect number, is that which is equal to the sum of 
all its aliquot parts : thus 6=f-|-f +§-. 

Prop. 1. — The sum or difference of any two even numbers 
is an even number. 

2. The sum or difference of any two odd numbers is even ; 
but the sum of three odd numbers is odd. 

3. The sum of any even number of odd numbers is even ; but 
the sum of any odd number of odd numbers is odd. 

4. The sum or difference of an even and an odd number is 
odd. 

5. The product of an even or an odd number, or of two even 
numbers, is even. 

6. An odd number cannot be divided by an even number, 
without a remainder. 

7. Any power of an even number is even. 

8. The product of any two odd numbers is an odd number. 

9. The product of any number of odd numbers is odd ; and 
every power of an odd number is odd. 

10. If an odd number divides an even number, it will also 
divide the half of it. 



PROPERTIES OF NUMBERS. 51 

11. If a number consist of many parts, and each of those parts 
have a common divisor d, then will the whole number, taken 
collectively, be divisible by d. 

12. Neither the sum nor the difference of two fractions, 
which are in their lowest terms, and of which the denominator 
of the one contains a factor not common to the other, can be 
equal to an integer number. 

13. If a square number be either multiplied or divided by a 
square, the product or quotient is a square ; and conversely, 
if a square number be either multiplied or divided by a 
number that is not a square, the product or quotient is not a 
square. 

14. The product arising from two different prime numbers 
cannot be a square number. 

15. The product of no two different numbers prime to each 
other can make a square, unless each of those numbers be a 
square. 

1 6. The square root of an integer number, that is not a com- 
plete square, can neither be expressed by an integer nor by any 
rational fraction. 

17. The cube root of an integer that is not a complete 
cube cannot be expressed by either an integer or a rational 
fraction. 

18. Every prime number greater than 2, is of one of the 
forms 4 n + 1, or 4 n — 1. 

19. Every prime number greater than 3, is of one of the 
forms 6 n + 1, or 6 n — 1. 

20. No algebraical formula can contain prime numbers 
only. 

21. The number of prime numbers is unlimited. 

22. The first twenty prime numbers are 1, 2, 3, 5, 7, 11, 13, 
17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, and 67. 

23. A square number cannot terminate with an odd number 
of ciphers. 

24. If a square number terminate with a 4, the last figure 
but one (towards the right hand) will be an even number. 

25. If a square number terminate with 5, it will terminate 
with 25. 

26. If a square number terminate with an odd digit, the last 
figure but one will be even ; and if it terminate with any even 
digit, except 4, the last figure but one will be odd. 

27. No square number can terminate with two equal digits, 
except two ciphers or two fours. 

28. No number whose last, or right hand, digit is 2, 3, 7, or 
8, is a square number. 



52 PROPERTIES OF NUMBERS. 

29. If a cube number be divisible by 7, it is also divisible by 
the cube of 7. 

30. The difference between any integral cube and its root is 
always divisible by 6. 

31. Neither the sum nor the difference of two cubes can be a 
cube. 

32. A cube number may end with any of the natural num- 
bers 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0. 

33. If any series of numbers, beginning from 1, be in con- 
tinued geometrical proportion, the 3d, 5th, 7th, &c. will be 
squares ; the 4th, 7th, 10th, &c. cubes ; and the 7th, of course, 
both a square and a cube. 

34. All the powers of any number that end with either 5 or 
6, will end with 5 or 6, respectively. 

35. Any power, n, of the natural numbers, 1, 2, 3, 4, 5, 6, 
&c. has as many orders of differences as there are units in the 
common exponent of all the numbers ; and the last of those 
differences is a constant quantity, and equal to the continual 

product Ix2x3x4x X n, continued till the last 

factor, or the number of factors be n, the exponent of the powers. 
Thus, 

The 1st powers 1, 2, 3, 4, 5, &c. have but one order of 

differences 1111 &c. and that difference is 1. 
The 2d powers 1, 4, 9, 16, 25, &c. have two orders of 
differences 3 5 7 9 
2 2 2 
of which the last is constantly 2=1 X 2. 
The 3d powers 1, 8, 27, 64, 125, &c. have three orders of 
differences 7 19 37 61 
12 18 24 
6 6 
of which the last is 6 = 1 x 2 x 3. 

In like manner, the 4th, or last, differences of the 4th powers, 
are each = 24= 1 x 2x3x4; and the 5th, or last 
differences of the 5th powers, are each 125 = 1 x 2 x 3 x 
4x5. 

36. If unity be divided into any two unequal parts, the sum 
of one of those parts added to the square of the other, is equal to 
the sum of the other part added to the square of that. Thus, 
of the two parts | and «, l + HY = 1 + (i) 2 = U > so > a S ain > 
of the parts f and |, § + (|) 2 = | + (f ) 2 = flf. 

For the demonstrations of these and a variety of other pro- 
perties of numbers, those who wish to pursue this curious line 



PROPERTIES OF NUMBERS. 53 

of inquiry may consult Legendre " Sur la Theorie des Nom- 
bres," the " Disquisitiones Arithmeticae" of Gauss, or Barlow's 
" Elementary Investigation of the Theory of Numbers." 

Also, for the highly interesting properties of Circulating 
Decimals, and their connexion with prime numbers, consult 
the curious works of the late Mr. H. Goodwyn, entitled "A 
First Centenary," and "A Table of the Circles arising from the 
Division of a Unit by all the Integers from 1 to 1024." 

Ji useful Numerical Problem, to reduce a given fraction or 
a given ratio, to the least terms ; and as near as may be 
of the same value. 

Rule 1. — Let A, B, be the two numbers. Divide the latter 
B by the former A, and you will have 1 for A ; and some num- 
ber and a fraction annexed, for B, call this C. Place these in 
the first step. 

Then subtract the fractional parts from the denominator, and 
what remains put after C + 1, with a negative sign. Then 
throw away the denominator, and place 1 and that last number 
in the second step. This is the foundation of all the rest. 

If the fractional parts in both be nearly equal, add these two 
steps together ; if not, multiply the less by such a number 
as will make the fractional parts, in both, nearly equal, and 
then add. And this multiplier is found by dividing the greater 
fraction by the less, so far as to get an integer quotient. When 
you add the steps together, you must subtract the fractional 
parts from one another, because they have contrary signs. 

The process is to be continued on, the same way, adding the 
last step, or its multiple, to a foregoing step, viz. to that which 
has the least fraction. 

Note. — The ratios thus found will be alternately greater and 
less than the true one, but continually approaching nearer 
and nearer. And that is the nearest in small numbers, which 
precedes far larger numbers : and the excess or defect of any 
one is visible in the operation. 



54 



DETERMINATION OP RATIOS. 



Example 1. 
To find the ratio of 10000 to 7854, in small numbers. 



9 
10 
11 

12 

13 

14 

15 



A B 

1 + -7854 

1 i_.2i46, 1st ratio. 

•2146 )-7854( 3 
3 3— -6438 



4 

5 

9 

14 

210 



•0044 



3 + -141 6, 2d ratio. 

4— -0730, 3d ratio. 

7 + -0686, 4th ratio. 
11 — -0044, 5th ratio. 
•06S6 ( 15 
165— -0660 



219 172 + -0026, 6thratio. 

233 1 8 3— -00 1 8, 7th ratio. 

452 355 -f. -0008, 8th ratio. 

•0008) -0018 (2 
904 710 + -0016 



1137 893— -0002, 9th ratio. 

•0002 ) -0008 ( 4 
4548 3572— -0008 



5000 



3927±*0000, 10th ratio. 



Explanation. 



The ratio of 10000 to 7854 is the same as 1 to + -7854 or 
1 to 1 — -2146 ; here 1 and 1 is the first ratio. But 2146 be- 
ing less than 7854, divide the latter by the former, and you get 
3 in the quotient, then multiply 1 and 1 — -2146 by 3, pro- 
duces 3 and 3 — -6438 as in the 3d step. This third step added 
to the first step produces 4 and 3 for the integers, and subtract- 
ing the fractional parts, leaves *1416. So the 4th step is 4 and 
3 4- *1416 ; and the integers 4 and 3 is the 2d ratio. In this 
manner it is continued to the end ; and the several ratios ap- 
proximating nearer and nearer are T , |, f , f , \\, f If, fff , ff#> 
^y/ and ||4t- Here \\ is the nearest in small numbers, the 
defect being only y^*. 



DETERMINATION OP RATIOS. 



55 



Example 2. 

To find the ratio of 268*8 to 282 in the least numbers. 

2688 ) 2820 ( l£&=2— |f ff. 

2688 



132 

1 1+0132, first ratio. 

1 2—2556 

132)2556(19 
19 19 + 2508 



20 21— 48, 2d ratio. 

48 ) 132 ( 2 
40 42— 96, 



41 43+ 36, 3d ratio. 

61 64— 12, 4th ratio. 

12 ) 36 ( 3 

183 192— 36 



224 



235 , 5th ratio. 



So the several ratios are £,§-£, ||, fi, fff And the de- 
fect or excess is plain by inspection, e. g. ±±. differs from 
the truth only ^ff T P arts 5 an( * fr> Dut 48 suc ^ parts. 

Rule 2. — Divide the greater number by the less, and the 
divisor by the remainder, and the last divisor by the last re- 
mainder, and so on until remains. Then 

1 divided by the first quotient, gives the first ratio. 

And the terms of the first ratio multiplied by the second 
quotient, and 1 added to the denominator, give the second 
ratio. 

And in general the terms of any ratio, multiplied by the next 
quotient, and the terms of the foregoing ratio added, give the 
next succeeding ratio. 



56 DETERMINATION OP RATIOS. 

Example 3. 

Let the numbers be 10000 and 31416, or the ratio -i^f, 
10000 ) 31416 ( 3 
30000 



1416 ) 10000 ( 7 
9912 



88) 1416(16 
88 

536 

528 

8)88(11 
88 



Then ■£= 1st or least ratio. 

lx7 or ^r= 2d ratio. 

3x7+1 or 22 

7Xi6+ i_ or H2. = 3d ratio. 

22X16+3 OI 355 

113XU+ ?_ nr 1*5?= 4th ratio. 

355X11 + 22 ° r 3927 

Example 4. 

The ratio of 268-8 to 282 is required. 
2688 ) 2820 ( 1 
2688 

132)2688(20 
264 

48 ) 132 ( 2 
96 

36)48(1 
36 

12 ) 36 ( 3 
36 



DETERMINATION OF RATIOS. «S 7 

Then \ = first ratio. 

1X20 20 i ,. 

or — == 2d ratio. 



1X20 + 1 21 

20x2+1 41 

21^2+1 ° r 43 = 3d rat10 ' 

41 X1+20 61 ..- 

43x1+21 or ^ = 4th ratio. 

61x3+41 224 

64X" 3+43 0r 235 = 5th ratl °* 



58 ALGEBRA : DEFINITIONS. 



CHAPTRE II. 

ALGEBRA. 
Section I. — Definitions and Notation. 

Jllgebra is the science of the computation of magnitudes in 
general, as arithmetic is the particular science of the computa- 
tion of numbers. 

Every figure or arithmetical character has a determinate 
and individual value ; the figure 5, for example, represents 
always one and the same number, namely, the collection of 
5 units, of an order depending upon the position and use of 
the figure itself. Algebraical characters, on the contrary, 
must be, in general, independent of all particular significa- 
tion, and proper to represent all sorts of numbers or quan- 
tities, according to the nature of the questions to which we 
apply them. They should, moreover, be simple and easy to 
trace, so as to fatigue neither the attention nor the memory. 
These advantages are obtained by employing the letters of 
the alphabet, a, b, c, &c. to represent any kinds of magnitudes 
which becomes the subjects of mathematical research. The 
consequence is that when we have resolved by a single alge- 
braical computation all the problems of the same kind proposed 
in the utmost generality of which they are susceptible ; the 
application of the investigation to all particular cases requires 
no more than arithmetical operations. 

It is usual, though by no means absolutely necessary, to re- 
present quantities that are known by the commencing letters 
of the alphabet, as a, b, c, d, &c. and those that are unknown 
by the concluding letters, w, x, y, z. But it is often conve- 
nient, especially as it assists the memory, to represent any quan- 
tity, whether known or unknown, which enters an investiga- 
tion, by its initial letter, as sum by s, product by p, density 
by d, velocity by v, time by t, and so of others. 

Now, if s denotes the sum of four numbers represented by a, 
b, c, and d, then, adopting the other symbols explained at 
the beginning of arithmetic, we should express this algebrai- 
cally by writing s-=a-\-b+c-\-d. 



ALGEBRA I NOTATION. 59 

If the four quantities be all equal, ovs— a-\-a-\-a-\-a, 
this evidently reduces to s = 4 x a, or simply s — 4 a, drop- 
ping the sign of multiplication, which is here understood. 
The figure 4 is named the coefficient. In the quantities 3 a, 5 
a, 7 a,n a ; 3, 5, 7, and n, are respectively the coefficients. 

The continual product of three or more quantities is expressed 
either by interposing the sign of multiplication, as a x b x c 
X d ; or by interposing dots, which have the same signification, 
as a . b . c . d; or, lastly, by placing the letters in juxtaposition, 
as a b c d. 

When the quantities are equal, their continued multiplication 
produces powers, as a a, a a a, a a a a, &c. which are usually 
represented, instead of repeating the letters, by placing a figure 
a little above the single letter, to expound or tell how many 
equal factors are multiplied together ; this figure is called the 
exponent. Thus, instead of a a, a a a, a a a a, we put a 2 , a 3 , 
a 4 , the figures 2, 3, 4, being the exponents. 

Since roots are the reverse of powers, they are expressed by 
exponents, which are the reciprocals of those that express the 
corresponding powers. Thus the square root of a is represented 

either by s/a, or by « 2 ; the cube root of a + b, either by 

-tya + b, or by (a + b) 3 ; the fourth root of a + b — c, 

either by ^/ (a + b — c), or by {a + b — c) 4 . 

We give the name term to any quantity separated from 
another by the sign + or — . A monomial has one term ; a 
binomial has two terms, as a + b, a c — 4 a b ; when the 
second term of a binomial is — , it is frequently called a resi- 
dual. A trinomial has three terms, as«-f& + c, ad — 4ab 
+ 5 b c. A quadrinomial has four, asa + ^ + c — d. A 
multinomial, or polynomial, has many terms. 

The signs + and — , which in arithmetic simply indicate 
the operations of addition and subtraction, are employed 
more extensively in algebra, to denote, besides addition and 
subtraction, any two operations or any two states which are 
as opposed in their nature as addition and subtraction are. 
And if, in an algebraical process, the sign + is prefixed to a 
quantity to mark that it exists in a certain state, position, 
direction, &c, then, whenever the sign — occurs in connex- 
ion with such quantity, it must indicate precisely the con- 
trary state, position, &c. and no intermediate one. This is a 
matter of pure convention, and not of metaphysical reasoning. 

Other characters might have been contrived to denote this 
opposition ; but they would be superfluous, because the cha 



60 ALGEBRA : NOTATION. 

racters + and — , though originally restricted to denote 
addition and subtraction, may safely be extended to other 
purposes. 

Thus if -f a C added above } . C subtracted below, 

signifies any < to the right, > fi 1 °" < to the left, 
thing ( forwards, j £ backwards. 



— a signi- 
fies a cor- 
respond- 
ing 



" Decrease, 
Levity, 
Money owing, 
Motion downward. 



jc . . f Increase, 

nifcsTrfi Gravit y> 

71 Money due, 

® (^ Motion upwards, - 

And so on in every kind of contrariety. And two such 
quantities connected together in any case destroy each other's 
effect, or are equal to nothing, as + a — a = 0. Thus, if a 
man has but 10/. and at the same time owes 10/. he is worth 
nothing. And, if a vessel which would, otherwise, sail six 
miles an hour, be carried back six miles an hour by a current, 
it makes no advance. 

Like signs are either all positive ( 4- )? or a ^ negative ( — ). 
And unlike are when some are positive and others negative. 
If there be no sign before a quantity, the sign -|- is under- 
stood. • 

An equation is when two sets of quantities which make an 
equal aggregate are placed with the sign of equality ( = ) be- 
tween them : 

As 12 + 5 = 20 — 3, or x + y — a -\- b = c d. 

The quantities placed on both sides the sign of equality are 
called respectively the members of the equation. 



Section II. — Addition and Subtraction. 

1. Properly speaking, there is not in algebra either addi- 
tion or subtraction, but a reduction, namely, the algebraic 
operation, by which several terms are, when it is possible, 
combined into one term. This, however, can only be effected 
upon quantities that differ in their coefficients and their 
signs, while they are formed of the same letters and the same 
exponents. 

Thus, 3 «, + 4a, + 7 «, § ^ J are eyidently reducible . 



ALGEBRA I ADDITION AND SUBTRACTION. 



61 



In the first set, the incorporation gives (3+4 + 7) a— 14 a, 
m the second (5 — 3 + 8) a 3 b=lO a 3 b. 

2. Generally, taking the similar terms the reduction affects 
their coefficients, which are to be added when their signs are 
alike, subtracted when they are different : in the first case, 
give to the result the common sign ; in the second, the sign 
of the quantity having the greatest coefficient. 

3. When quantities are presented promiscuously, it is best to 
classify them, previously to the incorporation. 



Thus, 3 a 2 , — 3 b c, + 2 c 2 , + 
4d, + 7 a 3 , + 5 b c, + a 2 ,— 2 c 2 , 
— 4 be, when arranged become 
as in the margin, and their sum 
is readily obtained, as in the fourth 
line. 



3 a 2 + 5bc — 2 c z 

7a 2 — 3bc + 4d 

a 2 — 4 b c + 2 c 3 

11 a 3 — 2b c-\- 4d 



4. If it were required to subtract a residual, as b — c, from 
a single term, as a ; it is evident that the required difference 
would not be changed if a quantity c were added to both. We 
should then have to take b — c + c, or b, from a + c, that is, 
we should, have a + c — b, for the difference sought, in which, 
as is manifest, the signs of the letters b and c, which were to be 
subtracted, have become changed. 

Hence, to subtract a polynomial, change all the signs, and 
reduce by incorporating the coefficients, where that is pos- 
sible. 



Thus, 4 a b — 3b c And 

-—(2 ab — Qbc) 



become 

Result 2 a b + 3b c Result 



4 a b — 3b c become 
2 ab + §bc 



4ab — 3c 2 + be 
■{ab— c*-2bc) 



4ab — 
—ab + 


3c 3 + be 
c 3 +2 b c 


3ab + 


2c 2 +3bc 



5. In addition and subtraction of algebraic fractions, the 
quantities must be reduced to a common denominator, and 
occasionally undergo other reductions similar to those in 
vulgar fractions in arithmetic : and then the sum of the dif- 
ference of the numerators may be plaeed over the common 
denominator, as required. 

10 G 



62 ALGEBRA .* ADDITION AND SUBTRACTION. 

rp. a <c a d .b c ad-\-bc 

b d b d b d b d 

A , a . b , c . ab , ac , be a 2 
And, — + — +— 7-+ -\ 



+ 



be a c ab c b a a be 

b 2 c 2 a 2 b 2 a 2 c 2 b 2 c 2 



abc abc abc abc abc 
a 2 +b 2 +c 2 +a 2 b 2 +a 2 c 2 +b 2 c 2 
abc 

A1 a c ad — be 
Also, - — = 



And, 



b d bd ' 

a+x a — x ac+cx ad — dx 



d c c d d c 

ac+cx — ad+dx a (c — d)+x(c+d) 
cd cd 

b-\-x h — x 



And, 



b—x b -\- x 



(b 2 + 2 bx+x 2 )— {b 2 — 2 bx+x*)__ 4bx 
& — x 2 b 2 — x 2 



Section III. — Multiplication. 

1. To multiply monomials, multiply their coefficients, add 
together the exponents which affect the same letters (ascrib- 
ing the exponent 1 to quantities which have none), then write 
in order the coefficients and letters thus obtained. 

2. To find the product of two polynomials, multiply each 
term of the one into all those of the other, following the rule 
given for monomials : and observe to take each partial product 
negatively when its factors have contrary signs, and positively 
when they have the same signs. Or, briefly, observe that like 
signs give +, unlike signs — . 

3. To multiply algebraic fractions, take the product of the 
numerators for the new numerator, and that of the denomina- 
tors for the new denominator. 

Note. — The general rule for the signs may be rendered 
evident from the following definition : multiplication is the 



Algebra: multiplication. 63 

finding a magnitude which has to the multiplicand the pro- 
portion of the multiplier to unity. Hence, the multiplier 
must be an abstract number, and, if a simple term, can have 
neither + nor — prefixed to its notation. Now 1st, + a x 
4. m = + m a, for the quality of a cannot be altered by in- 
creasing or diminishing its value in any proportion ; therefore 
the product is of the quality plus, and m a by the definition is 
the product of a and m. Secondly, — ax + m = — ma, 
for the same reasons as before, mutatis mutandis. Thirdly, 
4- a X — m> has no meaning ; for m must be an abstract 
number, therefore here we can have no proof. But + a x 
(m — n) = m a — n a, n being less than m ; for a taken as 
often as there are units in m is = m a by the first case ; but a 
was to have been taken only as often as there are units in 
m — n ; therefore a has been taken too often by the units in n ; 
consequently a taken n times or n a, must be subtracted ; 
and of course m a — n a is the true product. Fourthly, 
— a x (m — n) — — ma + n a. For — a x m — — ma 
(by case 2) ; but this, as above, is too great by — n a ; there- 
fore — ma with n a subtracted from it is the true product ; 
but this, by the rule of subtraction, is = — m a + n a. 



Examples. 

1. 4 a b x 5cd=4.5.ab.cd = 20abcd. 

2. 8 a* b 3 x 4 a 5 b = 8 . 4 . a 2 . a 5 . b 3 . b = 32 a a + 5 
b 3 + 1 = 32 a 7 b\ 



Multiply 2 a + b c — 2 b* 
By 2 a — b c + 2 b* 



4a» + 2abc — 4ab 2 

— 2abc — b*c a + 2b 3 c 

+ 4 a b* + 2 b 3 c — 4 b 4 



Product 4 a 9 — 


-b* 


c 3 + 4 b 3 c — 4b 4 


a -f b 
a + b 


5. 


(a* + 2 ab +b» 
\a + b 


a* + a b 

ab+b* 


a 3 + 2 a*b + a b* 

+ a* b + 2 a b» + b 3 


a' + 2ab +b» 


a 3 + 3 a 2 b + 3 a b» + b* 



64 ALGEBRA : DIVISION 

la — o 



a* + a b 
— ab — b* 

a 2 — b 2 



a + b a — b {a -\. b){a — b) a 2 -—b 2 

c d c x d cd 

2x 3ab 3ac 18 a 2 bcx 9 ax 

8 '~ X ~~7~ X TV "" 2abc ** ~T~ X 

2 a b c 

= — — 9 a x. 

2 a b c 

Note. — From the above examples (4, 5, and 6) we may- 
learn — 

1. That the square of the sum of two quantities is equal to 
the sum of the squares of the two quantities together with twice 
their product. 

2. That the product of the sum and difference of two quan- 
tities is equal to the difference of their squares. 

3. That the cube of the binomial a + b, is a 3 + 3 a 2 b + 
3ab 2 + b\ 



Section IV. — Division. 



1. To divide one monomial by another, suppress the letters 
that are common to both, subtract the exponents which affect 
the same letters, and divide the coefficients one by another. 

2. To divide a polynomial by a monomial, divide each term 
of the polynomial by the monomial according to the rule 1, and 
connect the results by their proper signs. 

3. To divide two polynomials one by the other, arrange them 
with respect to the same letter, then divide the first terms one 
by the other, and thence will result one term of the quotient ; 
multiply the divisor by this, and subtract the product from the 
dividend : proceed with the remainder in the same manner. 
Observe in the partial divisions the same rules for the determi- 
nation of the signs as in multiplication. 



ALGEBRA I DIVISION. 65 

4. To divide algebraic fractions, invert the terms of the 
divisor, and proceed as in multiplication. 



Examples 

1. 12 a 3 b»c-r- 3 ab= if a 3 - 1 ^- 1 c = 4 a* b c. 

2. 15a 3 b 5 -r-5 a*b* = if a 3 ~ 2 b s - z = 3 a b 3 . 

6 x 2 12 rw 

3. 6 x* + 12xy — 9 x y z -i- 3x = 4- y _- 

9 9 3x T 3a? 

4. Divide x 3 — 3 x z z + 3 x z* — z 3 by x — z. 

x — z)x 3 — 3x»z + 3x ^ — ^(a? 2 — 2xz+ z* quotient. 
x 3 — x*z 



— 2 x* z + 3 x z 2 

— 2x*z + 2xz* 



xz^ — z 5 

X z* — z 3 



5. Divide a 5 — b s by a — b. 
a — b)a s — b 5 {a 4 + a 3 b + a* b 9 + ab 3 + b 4 quotient. 
a* — a 4 b 



a 4 b 

a 4 b — a 3 b* 

a 3 b* 

a 3 b* — a 9 b 3 

Here the second term of a 2 b 3 

the dividend is brought a 3 b 3 — a b 4 

down to stand over the 

corresponding term in ab 4 — b 5 

the last product. ab 4 — b 5 



a2 



66 algebra: division. 

6. Divide 1 by 1 — x. 

1— x)l (l+x + x 2 + x 3 + x 4 +— 
l—x 1 ~ 



x 



2x* 



X 
X- 


-X* 


-X 3 


-X* 






X 2 
X 2 - 








X 3 
X 3 - 






2x 2 


X* 
X 4 - 


'X 8 






X s 


X 


a 


+ x 



a 3 -f X s ' a + x a 3 + a? 3 x (a 3 + x 3 ) x 

2x 



x 2 — ax + a» 

x* — b 4 * x* + bx x 4 — b 4 x—b 



8 



x 9 — 2bx+b 2 ' x — b ~ (x — by x(x-\-b)' 

x* — b 4 ot — b 4 a» + b 2 b» 

x+ —. 



x (x + b) (x — b) x (x 2 — b 2 ) x x 

9. Divide 96 — 6 a 4 by 6 — 3 a. Quot. 16 -f 8 a + 4 a a 
+ 2 a 3 . 

10. Divide 10 a 3 + 11 a*b — 19 ab c — 15 a 2 c + 3a b 2 + 
15&C 9 — 5 6 a c by 3fflH5fl a —55 c. Qwo*. 2a + b — 3c. 

11. Divide s 9 + y a + -|- a by x + y + — . Quot. x — y 
V 2 



ALGEBRA I INVOLUTION. 67 

Section V. — Involution. 
1. — To involve or raise monomials to any proposed power. 

Involve the coefficient to the power required, for a new co- 
efficient. Multiply the index of each letter by the index of the 
required power. Place each product over its respective letter, 
and prefix the coefficient found as above : the result will be the 
power required. 

All the powers of an affirmative quantity will be -f; of a 
negative quantity, the even powers, as the 2d, 4th, 6th, &c. 
will be + ; the odd powers, as the 3d, 5th, 7th, &c. will 
be—. 

To involve fractions, apply these rules to both numerator and 
denominator. 

Examples. 

1. Find the fourth power of 2 a 3 . 
2x2x2x2 = 16, new coefficient. 

2 x4=8, new exponent. Hence 16 a 8 the answer. 

2. The fifth power of — 3 y 3 is 243 y 10 . 

3. The fourth power of — 4 a; 3 is 256 x w . 

2 z 64 z e 

4. The sixth power of — 2 is ^—f 2 

2. To involve polynomials. 

Multiply the given quantity into itself as many times, want- 
ing one, as there are units in the index of the required power, 
and the last product will be the power required. 



Cube x ± z and 2 x - 
x db z 
x ± z 


Exc 

-3 z. 

ubes. 


tmple. 

2x—3 z 
2x — 3z 




x»±xz 
rh X Z+Z* 


4 x 2 — 6xz 

— 6xz+9z» 




a^± 2 xz+z 2 squares, 
x ±z 


4x*—12xz+9z* 
2x — 3 z 




ar*±2 x z z+xz 2 
db x* z+2xz*+z* 


8x*—24x 2 z+l8xz* 
— 12x*z+36xz*- 


-27 s 3 


x*±:3x*z+3xz 2 +z 3 c 


Sx 3 — 36a; 3 2'+54a?ir 3 - 


-27r* 



68 ALGEBRA I INVOLUTION. 

The operation required by the preceding rules, however 
simple in its nature, becomes tedious when even a binomial 
is raised to a high power. In such cases it is usual to employ 

Sir Isaac Newton's Rule for involving a Binomial. 

1. To find the terms without the coefficients. — The index 
of the first, or leading quantity, begins with that of the given 
power, and decreases continually by 1, in every term to the 
last ; and in the following quantity the indices of the terms are 
0, 1, 2, 3, 4, &c. 

2. To find the uncise or coefficient. — The first is always 1, 
and the second is the index of the power : and in general, if 
the coefficient of any term be multiplied by the index of the 
leading quantity, and the product be divided by the number of 
terms to that place, it will give the coefficient of the term next 
following.* 

Note. — The whole number of terms will be one more than 
the index of the given power ; and when both terms of the root 
are 4- , all the terms of the power will be + ; but if the second 
term be — , all the odd terms will be +> and the even terms — . 

Examples. 

1. Let a -f x be involved to the fifth power. 

The terms without the coefficients will be 

a 5 , a 4 x, a 3 x 2 , a* X s , a x 4 , x s , 

and the coefficients will be 
5x4 10x3 10X2 5X1 



1,5, 



IT' ~^~' ~T~' -5— ' 



And therefore the fifth power is 
a 5 +5a 4 x+10 a 3 x 2 +\0 a 2 x 3 -\-5 a x 4 +x 5 . 

* This rule, expressed in general terms, is as follows : 
(a+by =an+n.a»-i b -f n . n ~ 1 &**& +n.!liJ.. 

n ~~ 2 .q"-3 63,&c. 
3 

(a-J)"^"-?!.^- 1 b + n. r LH±a*-*b*— n. n Zl. 

2 2 * 

3 

The same theorem applied to fractional exponents, and with a slight modifica- 
tion, serves for the extraction of roots in infinite series ; as will be shown a little 
farther on. 



ALGEBRA I BINOMIAL RULE. 69 

Here we have, for the sake of perspicuity, exhibited sepa- 
rately, the manner of obtaining the several terms and their re- 
spective coefficients. But in practice the separation of the two 
operations is inconvenient. The best way to obtain the coeffi- 
cients is to perform the division first, upon either the requisite 
coefficient or exponent (one or other of which may always be 
divided without a remainder), and to multiply the quotient into 
the other. Thus, the result may be obtained at once in a single 
line, nearly as rapidly as it can be written down. 

2. (x + y) 7 =x 7 + 1 x 6z + 2lx 5 z 2 + 35x 4 z 3 + 35 x*z* 
+ 21 x 2 z s + 7 x z 6 + z 7 . 

3. {x — zf = x 8 — 8 x 7 z + 28 x 6 z 2 — 56 x 5 z 3 + 70 x 4 z 4 
— 56 x z z 5 + 28x 2 z 6 — 8xz 7 + z 8 . 

For Trinomials and Quadrinomials. — Let two of the terms 
be regarded as one, and the remaining term or terms as the 
other ; and proceed as above. 



Example. 

Involve x + y — z to the fourth power. 

Let x be regarded as one term of the binomial, and y — z as 
the other : then will [x + y — z) 4 = {x + (y — z)} 4 = x 4 4- 
4x 3 (y—z) + 6 x 2 (y — z) 2 + 4x (y — zf + (y — z) 4 , 
where the powers of y — z being expanded by the same rule, 
and multiplied into their respective factors, we shall at length 
have x 4 -f 4 x 3 y — 4 x 3 z -f 6x* y 2 — 12 x 2 y z + 6 x 2 z 2 + 
4 x y 3 — 12 x y 2 z -\- 12 x y z 2 — 4 x z 3 + y 4 — 4y B z -\- 6 
y 2 z 2 — Ay z 3 + z 4 , the fourth power required. 

Had (x + y) and — z been taken for the two terms of the 
binomial, the result would have been the same. 

Note. — The rule for the involution of multinomials is too 
complex for this place. 

11 



70 ALGEBRA I EVOLUTION. 



Section VI. — Evolution. 



1 . To find the roots of monomials. — Extract the correspond- 
ing root of the coefficient for the new coefficient : then mul- 
tiply the index of the letter or letters by the index of the root, 
the result will be the exponents of the letter or letters to be 
placed after the coefficient for the root required. 



Examples. 

1. Find the fourth root of 81 a 4 z\ 
First *y 81 = <s/9=3, new coefficient. 
Then 4 X i = 1, exponent of a, and 8 X £ = 2, expo- 
nent of z. 
Hence 3 a z 2 is the root required. 

2. ^/(32 a s x 10 ) = */32 xa sX ? x* 10x *=2 a x* 

3 \ 8x3 _ ^ 8 X ^ 3 X * 2 x 
3 - ^27z 9 ~V27xz 9 ~* =Z3z3 ' 

2. To find the square root of a polynomial. — Proceed as 
in the extraction of the square root, in arithmetic. 



Examples. 
1. Extract the square root of a 4 -f 4 a? x + 6 a 3 x* + 4 ax 9 

+X 4 . 

a 4 +4a 3 a; + 6a*x 2 + 4 ax 3 + x 4 (a 9 + 2 <zar -f x 2 
a 4 [root req. 

2 a 9 -f 2 ax)4a 3 x+ 6 a 2 x 2 
4a?x + 4a 2 x 2 



2 a* + 4ax+ x 2 )2 a*x 2 + 4ax* +x 4 
2 a 2 x* + 4 a x 3 + x 4 



ALGEBRA I EVOLUTION. 71 

2. Extract the square root of x 4 — 2 a? 3 -f f x* — \ x + T V. 

x 1 

x 4 — 2 x 3 + | # 3 — - +-- (a? 3 — a? + i ratf. 

x 4 2 16 

2a? 3 --ar) — 2a- 3 + |a? 2 
— 2 x 3 + x z 



** 2 + 16 



3. 7b jfmc? Me roots of powers in general. — If they be not 
the roots of high powers that are required, the following rule 
may be employed : 

Find the root of the first term, and place it in the quotient. 
— Subtract its power, and bring down the second term for a 
dividend. — Involve the root, last found, to the next lowest 
power, and multiply it by the index of the given power for a 
divisor. — Divide the dividend by the divisor, and the quotient 
will be the next term of the root. — Involve the whole root, 
and subtract and divide as before ; and so on till the whole is 
finished. 



Examples. 

1. Find the cube root of a? 6 — 6 x 5 + 15 x 4 — 20 x* -f 15 a? 3 
— 6ar + 1. 

a? 6 — 6a? 5 +15a? 4 — 20x*+l5x*— 6x+l)x z — 2x 
x 6 [+1 root req. 

(a? B )»x3=3a? 4 ) — 6 x s 

x 6 — 6 x s + 12 0^ — 8 x* 



3 a^)3 x 4 



a? 6 — 6x s + I5x 4 — 20x*+ 15x 2 —6x + 1 



72 algebra: evolution. 

2. Find the 4th root of 16 a 4 — 96 a 3 x + 216 a 2 x 2 — 
216 a V + 81 # 4 . 

16 a 4 — 96 « 3 x + 216 a 3 # 2 — 216 a # 3 +81 #4 ( 2 a 
16 a 4 [ — 3 x, root sought. 

8 a 3 x 4 = 32 a 3 ) — 96 a s x 



I6a 4 — 96a 3 x + 216 a*x 2 ~2\6 a x* + 81a? 4 



Note. — In the higher roots proceed thus : 
For the biquadrate, extract the square root of the square root. 

sixth root cube root of the square root. 

eighth root, sq. rt. of the sq. rt. of the sq. rt. 



ninth root, the cube root of the cube root. 



Examples, however, of such high roots seldom occur in any 
practical inquiries. 



Section VII. — Surds. 



A Surd, or irrational quantity, is a quantity under a radical 
sign or fractional index, the root of which cannot be exactly ob- 
tained. (See Arith. Sect. 9. Evolution.) 

Surds, as well as other quantities, may be considered as 
either simple or compound, the first being monomials, as v/3, 

a*, {/a b*, the others polynomials, as >/3 -f v/5, &a -f y/b 
— v/c 3 , &c. 

Rational quantities may be expressed in the form of surds, 
and the operation, when effected, often diminishes subsequent 
labour. 



Reduction. 
1. To reduce surds into their simplest expressions. 

1. If the surd be not fractional, but consist of integers or in- 
tegral factors under the radical sign : 

Divide the given power by the greatest power, denoted by 
the index, contained therein, that measures it without re- 
mainder ; let the quotient be affected by the radical sign, and 



ALGEBRA I SURDS. 73 

have the root of the divisor prefixed as a coefficient, or connected 
by the sign x . 



Examples. 

1. ^/75 = s/(25 X 3) = </25 X >/3 = 5 >/3. 

2. f 448 = f (64 X 7) = f 64 X f 7 = 4 f 7. 

3. ^176 = ^/(16x11) = V16 X t/H =2 V 11 - 

4. ^/(8a? s — 12 a? 3 y) = v/4 ^ (2 a? — 3 y) = ^4 a? 9 x 

V(2a;-3i/) = 2 a: — 3y). 

5. ^108a; 3 3/ 4 = f(27a? 3 # 3 X 4 y) =^27 a? 3 y 3 X ^4y 

= 3a?y f 4 y. 

6. f (56a? 3 y + 8 a; 3 ) = f 8 a? 3 (7 # + 1) = f 8 a? 3 x 

f (7y + 1) =2^(7^ + 1). 

2. If the surd be fractional, it may be reduced to an equiva- 
lent integral one, thus : 

Multiply the numerator of the fraction under the radical 
sign, by that power of its denominator whose exponent is 
1 less than the exponent of the surd. Take the denominator 
from under the radical sign, and divide the coefficient (whether 
unity, number, or letter) by it, for a new coefficient to stand 
before the surd so reduced. 

Note. — This reduction saves the labour of actually dividing 
by an approximated root ; and will often enable the student 
to value any surd expressions by means of a table of roots of 
integers. 



Examples. 

1. v/| =^(}.!) = ^=n/JXn/3 = | V3. 

2. ^/i = •(* . 4) = </& = Vi-s X v/5 =4 a/5. 

= y/n. 
4. f #=*f (f . ff )= f ■*•§-§ = f Tl h X f 100 = i. f 100. 

|2a __ |/2a 25 x 2 \ _ [ 50 a x 2 f 1 

\5x "" A^ \5~T • 25 x 2 / ™ \ 125 x 3 J" \ 125~^3 

X ^50 a a? 2 = — f 50 a xK 

(16 [8.2 [8.2 9 1 8. 18 f 8 

b * Sis! = N9T9 = N9T9 V = N"729 N 739" 
X f 18 = |f 18. 

H 



74 ALGEBRA I SURDS. 

3. If the denominator of the fraction be a binomial or re- 
sidual, of which one or both terms are irrational and roots of 
squares : 

Then, multiply this fraction by another which shall have 
its numerator and denominator alike, and each to contain the 
same two quantities as the denominator of the given expression, 
but connected with a different sign. 

Note. — By means of this rule, since any fraction whose nu- 
merator and denominator are the same, is equal to unity, the 
quantity to be reduced assumes a new appearance without 
changing its value ; while the expression becomes freed from 
the surds in the denominator, because the product of the sum 
and difference of two quantities is equal to the difference of 
their squares. 



Examples. 



8 _ 8 \/£+j/? __ 8 (y/5 + y/3) = 

* v/5— -v/3 "~ y/5— y/3 * ^ + \/ 3 ~ 2 

4 (y/5 + y/3). 



2. 



3 



y/5 — y/ 2 _ 3 (y/5— y/2) _ 



\/5 -f y/2 "~ y/5 + y/2 ' y/5 — y/2 
y/5 — y/2. 

y/20 — y/12 __ y/20 — y/12 y/5 — y/3 __ y/100— 2y/60-f-y/36 

* y/5 + y/3 ~" y/5 + y/3 * y/5 — y/3 ~~ 5 — 3 



y/q b y/a b yV5— yV3 _ ^5—^3 , 

vV 5 + v^ 3 ~" v' 5 + vV3 • yV5— yV3 ~~ y/5— y/3 ^ ^ ° 

. (i •« b). 

Note 2. — Upon the same general principle any binomial or 
residual surd, as ^/A ± fy B may be rendered rational by 
taking Z/A*- 1 =f C (A W ~ 3 B) + v^ (A n ~ 3 B 3 ) =f ^ (A n ~ 4 B 3 ) 
+ &c. for a multiplier : where the upper signs must be taken 
with the upper, the lower with the lower, and the series con- 
tinued to n terms. 

Thus, the expression tya? — y/6 3 , multiplied by y/a 9 -+ 
vVa 6 b* + ¥a* b 6 + yV6 9 , gives the rational product a 3 — A 8 . 



ALGEBRA : SURDS. 75 

2. To reduce surds having different exponents to equiva- 
lent ones that have a common exponent. — Involve the powers 
reciprocally, according to each other's exponent, for new powers: 
and let the product of the exponents be the common exponent. 

Note. — Hence, rational quantities may be reduced to the form 
of any assigned root ; and roots with rational coefficients may 
be so reduced as to be brought entirely under the radical sign. 



Examples, 

i i 1.12 m i« n ■» 

\. a n and b m , become a n m or a mn and b m n or b m n . 

X X X»A JL X.k 2 

2. a 2 and b 3 , become a 2 3 or a 6 and b 2 2 or b 6 . 

x x 1 i 

3. 3 2 and 2% become 3 6 and 2% or V3 3 and ^/2 a or V 27 
and ^/4. 

4. («+*)% and {a— bf, become ij/ (a+b)* and ty{a - b) 3 . 

5. The rational quantity a 2 becomes y/a 4 , -Va 6 , i/a 8 9 or 

6. 4 a &5 b f becomes ^(4 af x ^5 6, ^64 « 3 x &5 6, or 
^320 « 3 b. 

These and other obvious reductions, which will at once sug- 
gest themselves, being effected, the operations of addition, sub- 
traction, &c. are so easily performed upon such surd quantities as 
usually occur, that it will merely suffice to present a few exam- 
ples without detailing rules. 



•Addition. 

Ex. 1. ^8+</18=V(4.2) + >/(9.2)=2 ^/2+3 y/2=*5y/2. 

2. Add together ^54, </x, and </£.. 

^•?;^."}f} The sum of these is 

3. v/27 a 4 x+ x/3 a 3 #= ^/(9 a 4 . 3 #) + (a 2 . 3 a?) =3 «V3 
x+a^/3 a?=(3 a 3 +«)\/3 a?. 

4. 8WHv / fl 6 i=(8WXv / i) + (^ 6 X^)=8flv / 6 



76 algebra: surds. 



Subtraction. 



Ex. 1. 2>/50— v /18=2 % /(25.2)— </(9.2)=2.5 x /2— 3y/2= 
(10—3)^/2=7^/2. 

I ^15=^15. 

3. ^|_^ A= ^(|.|)-_^(_ V | )== ^i4-_^^ = 
J^18— ^18= T V^18; 

4. -^250 a 3 # — ^16 a 3 x=*&(l25 a 3 .2 x) — &{Sa 3 .2x) 
=5 a ^2 x—2 a &2 x=3 a &2 x. 

5. ^/45 s 4 x— s/20 s 2 a?= x/(9 s 4 . 5 a?)— V(4 $ 2 x».5x)=: 
(3 s* — 2 5a;) y/5 x. 

Vc/ \a/ 1 \« c/ « \« c/ a \ac' - 



Multiplication. 

Ex. 1. ^18x5^4=5 ^(18.4)=5 <^ (4.2.9) =5^(8.9) = 
5.2 ^9=10 ^9. 

2. | •§ x *•*-*•* ^ft-*)-! •*» 

MtVIHI i/3fr-4*CT v^5-a ^35. 

2 1 2 + i _8_ . _3_ J. 1 

3. a 3 x« 4 =« 3 4 =a 12 12 = a 12 . 



i„, . x* , , J+* 



5 



4. (x+z)' z x{x+z)* = (x+zy ' * = (x+z) G . 

5. (W Vy) X(x— ^y)=x*—y. 

i i_ i + i. w + w 
7. z n X z m = z» m — z 



m n 



8. y/dx Va b= W 3 XV« 2 6 2 =V« 2 & a d 3 . 

9. s/a— y/b^l/3'X s/a+ s/b-</3= </{a*— b + </3.) 

10. a n xar n =a m x—=a m - n . 

a n 

11. ^/ — aX\/— a=s/as/— IXs/a^/— 1=«X- 1=— «. 

12. s/ — a X \/ — b = </ a s/ — I X s/ b s/ — 1 = 
v/« 5 x — 1 = — </a b. 



algebra: surds. 77 



Division. 



Ex. 1. ^1000h-2 ^4=| ^i^=2 ^250= 
2^(125.2) = 10^2. 

2. f ^|-|^!=f|^(!-4)=fi^#= 

2 4 2 — 4 —2 2 

3. xS-r-x* = X s 3 =x 3 = i-j_o; 3 . 

11 11 m—n 

5. (a; 3 — xd—b+d s/b)-±(x — y/b)=x+ s/b — d. 
ac — ad a ac — ad 

6. — 2l — x/(fl 2 ai-flf)H-^->/(a-a:)= 2b X 

2 6 fa 2 a; — a x 2 



lb \a 2 x — ax 2 . 7N , 



Involution. 



Ex. l.(| a 8 )2=|.|.« 3 3 =| ds^Va*. 

*V •(H)'-rfr VCtV^Wy-* v^=t^ V2. 

3. (3+ ^/5) 3 =5(3+ n/5) (3+ ^5)1 = 14 + 6 ^5. 

4. (a— ^/6) 3 =« 3 — 3 a a v/^ + 3 ab— b ^/b. 



Evolution. 

Ex. 1. v/10 3 = ^1000= x/(100. 10)= ^100 X v/10 
= 10 s/10. 

2. 3/81 a 4 y 5 *= V (81 a 4 y 4 -?/ 3 z)= 9 a 3 t/ 3 ^/y 3 z. 

3. v/(« 3 — 4 a s/b-\-b 2 ) — a — 2 \/&> the operation being 
performed as in the arithmetical extraction of the square 
root. 

Note. — The square root of a binomial or residual a^kb, or 
even of a trinomial or quadrinomial, may often be convenient- 
ly extracted thus :— Take d = s/(a 2 —b 2 ) ; then 

^(azkb)=J—^- ± 4 J a ~ . This is evident : for, if J— ^— ± 

^ — - — be squared, it will give a + s/(a 2 — d 2 ) or a -f- b, as 
12 H 2 



78 ALGEBRA : SURDS. 

it ought : and, in like manner, the square of .1— — - 



J- 



a ~ d , is a — V{a* — d 3 ), or a — b. 



Ex. 1. Find the square root of 3 + 2 ^/2. 
Here «=3, b=2, */2, d= s/(9 — 8)=1, 

And 1±±*_± \ a - d _ l 3 + l j. [3 - 1 M 

\ 2 ^\ 2 \ 2 \ 2 

2. Find the square root of 6 — 2 \/5. 

Here « =6, 6=2 >/5,tf=V(36 — 20)= ^16=4, 

la+d |a — tf 16+4 16 — 4 
And ^J-g \j-2-=%|-l \| -2~ = 

•" — Vf = v/5-1. 

3. Find the square root of 6 + ^8 — </\2 — x/24. 
Here «=6+ ^8, 6= ^12+ n/24, d= n/(6 + ^8) 3 

— (v/12+ v/24) 3 = v/(44 + 12 ^/8 — - 36 — 2 x/12 . 24) 

= <v/(44 — 36 + 12^/8—12 V8) = x/8. 

n a + d 6+2 */8 „ '' j a— d 

Conseq. _L_==_I_vL =3+^8, and — = 

6+^/8 — ^8 

2 ~ d - 

But {Ex. 1), v/(3+2 v/2)= v/(3+ n/8)=1+ ^2. 
Therefore the root required is 1+ y/2 — v/3.* 



Section VIII. — Simple Equations. 

An algebraic equation is an expression by which two quanti- 
ties called members (whether simple or compound), are indi- 
cated to be equal to each other, by means of the sign of equality 
=placed between them. 

In equations consisting of known and unknown quantities, 
when the unknown quantity is expressed by a simple power, 
as x, x 2 , x 3 , &c. they are called simple equations, generally ; 

* For the cube and higher roots of binomials, &c. the reader may consult the 
treatises on Algebra by Maclaurin, Emerson, Lacroix, Bonnycastle, J. R. Young, 
and Hine. 



ALGEBRA I SIMPLE EQUATIONS. 79 

and particularly, simple or pure quadratics, cubics, &c. accord- 
ing to the exponent of the unknown quantity. But when the 
unknown quantity appears in two or more different powers in 
the same equation, it is named an adfected equation. Thus 
x* = a + 15, is a simple quadratic equation : x 2 + ax = b, 
an adfected quadratic. 

It is the former class of equations that we shall first con- 
sider. 

The reduction of an equation consists in so managing its 
terms, that, at the end of the process, the unknown quantity 
may stand alone, and in its first power, on one side of the 
sign =, and known quantities, whether denoted by letters, or 
figures, on the other. Thus, what was previously unknown is 
now affirmed to be equal to the aggregate of the terms in the 
second number of the equation. 

" In general the unknown quantity is disengaged from 
the known ones, by performing upon both members the re- 
verse operations,"* to those indicated by the equation, what- 
ever they may be. Thus, 

If any known quantity be added to the unknown quantity, 
let it be subtracted from both members or sides of the 
equation. 

If any such quantity be subtracted, let it be added.t 

If the unknown quantity have a multiplier, let the equation be 
divided by it. 

If it be divided by any quantity, let that become the multi- 
plier. 

If any power of the unknown quantity be given, take the 
corresponding root. 

If any root of it be known, find the corresponding power. 

If the unknown quantity be found in the terms of a propor- 
tion {Arith. Sect. 10), let the respective products of the means 
and extremes constitute an equation ; and then apply the gene- 
ral principle, as above. 

* This simple direction, comprehending the seven or eight particular rules for 
the reduction of equations given by most writers on algebra, from the time of 
Newton down to the present day, is due to Dr. Hutton. It is obviously founded 
upon the mathematical axiom, that equal operations performed upon equal things 
produce equal results. 

f These two operations constitute what \% usually denominated transposition. 



80 ALGEBRA : SIMPLE EQUATIONS. 



X 



Examples. 

1. Given x — 3 + 5 = 9, to find x. 

First, by adding} , , - n , ,_ 

r» j. l. "li. -j 6 fwe have a? -f- 5 = 9 + 3 = 12. 
3 to both sides, 3 

Then by sub- 5 g _ g _ u ^ ■ _ ?< 

tracting 5. 3 

Otherwise in appearance only, not in effect, 

By transposing the 3, and changing its sign, x -f 5 = 9 + 3. 

By transposing the 5, and changing its sign, a? = 9 + 3 — 

5=7. 

2. Given 3 x -f 5 = 20, to find a\ 

First, by transposing the 5, 3 a? = 20 — 5= 15 
by dividing by 3, x — \ s = 5. 

3. Given - + d = 3 b — 2 c to find a?. 

a 

First, transposing d, — =36 — 2 c + d. 

Then, multiplying by a, a? = 3 a 6 — 2ac + a fl?. 

4. Given ^(3 x + 4) + 2 = 6, to find a?. 

First transposing the 2, ^(3 x -f 4) = 6 — 2 =4. 
Then, cubing, 3 x + 4 = 4 3 = 64. 
Then, transposing the 4, 3 a? = 64 — 4 = 60 
Lastly, dividing by 3, a? = 6 3 = 20. 

5. Given 4a x — 5b=3dx-\-4c, to find a?. 

First, transposing 5 6 and 3 g? #, 4 « a? — 3 d x — 5 b -{- 4 c. 
That is, by collecting the coefficients, (4 a — 3^) x 
= 5 b + 4 c. 

Therefore, by dividing by 4 a — 3 d, x = - — — -~. 

6. Given la; -f r x — |- a? = 3, to find x. 
Multiplying by 120 = > 

4 x 5 x 6, we have 5 +^ 4 * — 20 * ~ 36 °- 

That is, collecting the coefficients, 34 x = 360. 
Therefore, dividing by 34, a? = ^ = ^ = 10^. 

7. Given % x : a :: 5 b : 3 c, to find ar. 
Mult, means and extremes, % c x = 5 a b 9 

Dividing by £ c, x =5ab +%c = !°£* 



ALGEBRA I SIMPLE EQUATIONS. 81 



8. Given a + x = s/a* + 'a? </ (4 b 2 -f a? 3 ), to find x. 
First, by squaring, we have, a 2 -f 2 « x -f a: 2 = « 3 + 
a; ^/(4 6 3 + x\) 

Then striking out # 3 from both sides, 2 ax + x 2 = 
a? v/(4 6 3 + x 2 ) 

dividing by x, 2 a + x = v/(4 b 2 + a: 2 ) 
squaring, 4 a* -f- 4 « a; + x 2 = 4 6 3 + a: 3 
striking out a- and 5 _ 4 A2 _ 4 ffl2 

transposing 4 a 2 3 

dividing by 4 a, x — — ^— == «. 



9. Given \/c x — a c = b + >/a: — a, to find a:. 

First, dividing by \/x — a, we have s/c = —r-, : + 1 

transposing the 1, \/c — 1, or — - — = — 

inverting and transposing the fractions, ^ ^ "~ — = . _ x 

multiplying by b, \/(x — a) = /c __ 1 . 

squaring both sides, x — a = _ 

transposing a, x = a -f 



c — 2 ^/c -f 1 



10. Given 13 — x/3 x = <v/13 -f- 3 a:, to find a:. 

Jlns. x — 12. 

11. Given y + </4 + */ 3 = _L_^ to find y. 

12. Given £ (a: +1)+ J (3 + 3)=i(a: + 4) + 16, to find a?. 

tflns. x — 41. 

13. Given J?^ : s/(x— 1) :: 3 : 1, to find x. Jins. lj. 

14. Given (b 4 + a: 4 )"* = (a 3 + # 2 )* to find a?. 



Jins. x 



~ a\ 2 ' 



Extermination. 



When two or more unknown quantities occur in the con- 
sideration of an algebraical problem, they are determinable by 
a series of given independent equations. In order, however, 



82 ALGEBRA I EXTERMINATION. 

that specific and finite solutions may be obtained, this condition 
must be observed, that there be given as many independent 
equations as there are unknown quantities. For, if the 
number of independent equations be fewer than the unknown 
quantities, the question proposed will be susceptible of an 
indefinite number of solutions :* while, on the other hand, a 
greater number of independent equations than of unknown 
quantities, indicates the impossibility or the absurdity of the 
thing attempted. 

Where two unknown quantities are to be determined from 
two independent equations, one or other of the following rules 
may be employed. 

1. Find the value of one of the unknown letters in each of 
the given equations ; make those two values equal to one 
another in a third equation, and from thence deduce the value 
of the other unknown letter. This substituted for it in either 
of the former equations, will lead to the determination of the 
first unknown quantity. 

2. Find the value of either of the unknown quantities in one 
of the equations, and substitute this value for it in the other 
equation : so will the other unknown quantity become known, 
and then the first, as before. 

3. Or, after due reduction when requisite, multiply the first 
equation by the coefficient of one of the unknown quantities in 
the second equation, and the second equation by the coefficient 
of the same unknown quantity in the first equation : then the 
addition or subtraction of the resulting equations (according as 
the signs of the unknown quantity whose coefficients are now 
made equal, are unlike or like) will exterminate that unknown 
quantity, and lead to the determination of the other by former 
rules. 

Notes. The third rule is usually the most commodious and 
expeditious in practice. 

The same precepts may be applied, mutatis mutandis, to 
equations comprising three, four, or more unknown quantities 
and they often serve to depress equations, or reduce them from 
a higher to a lower degree. 

* This, though generally true, has one striking exception, namely, in the case 
of equations constituted partly of rational quantities and partly of quadratic surds ; 
where two unknown quantities are determinable by one equation, four unknown 
quantities by two equations ; and so on. 

Thus, if x -f- s/y= a -f y/b 
and z — y/v = c — y/d 
Then x = a, y = b, z =■ c,v = d. 



ALGEBRA : EXTERMINATION. 83 



Examples. 

1. Given 4 x 3 -f 3 y — 41, and 3 a? 3 — 4 y = 12, to find x 
andy. 

1st equa. X by 3, gives 12 a? 3 + 9 y =123. 
2d equa. X by 4, gives 12 a? 3 — 16 y — 48. 
The difference of these, 25 y — 75, whence y = 3. 
Then, from equa. 2d, 3 x 3 = 12 + Ay = 12 + 12 = 24. 
Whence dividing by 3, x 3 = 8, or x = 2. 



^a-. 2. Given a?+y + z=53, a?+2y + 3*= 105, and 
x + 3y + 4z = 134. 

1. a? + y + z = 53 

2. a?+ 2y + 3*= 105 

3. x + 3y + 4z = 134 

4. 1st equa. taken from 2d, gives y + 2 z = 52 

5. 2d equa. taken from 3d, y -f £ = 29 

6. 5th equa. taken from 4th, z = 23 

7. 6th equa. taken from 5th, y =6 

8. 5th equa. taken from 1st, x = 24. 



Ex. 3. Given x + y = a, x + z = b, y -\- z — c, to find 
a?, y, and r. 

1. x -\- y — a 

2. a? + z = 6 

3. y + 2- = c 

4. 1st + 2d + 3d, gives 2 x + 2 y + 2 z = a + b + c. 

5. Half equa. 4th gives x-\-y-\-z— \a-\-\b-\-\ct 

6. 3d equa. taken from 5th, gives x = \ a + \b — \c. 

7. 2d equa. taken from 5th, y = \ a — \b + \ c. 

8. 1st equa. taken from 5th, z = — \ a -\- \b + \c. 



Ex. 4. Given a x + b y — c, and a x + b' y = c', to find 
a? and y. 



„ c b' — be' . ac' — c a' 

Jlns. x — —r, -j—, and y = — - — ■ 

ab — ba ab — ba 



84 



ALGEBRA : EXTERMINATION. 



Ex. 5. Given a x + by -j- c z = d, a' x -f b' y -f c' z = d' 
a" x + b" y + c" z = eT to find x, y and z. 

Jlns r- d b ' c " — dc ' b "+ c d ' b "~ b d ' c " + b c ' d "—cb' d'\ 
ab' c" — a c' b"-j-c a' b" — b a' c"-\-b c' a" — c b' a" 

ad' c" — ac' d"+ca' d" — da' c"-\-dc' a" — c d' a'' 

y — . ! ! • 

a ab' c" — a c' b"+c a' b" — b a' c"-\-b c' a" — c b' a" 

_ ab' d" — ad'b"-\-d a' b" — b a' d" -\-b d' a" — db' a" 
~~ a b' c" — a c' b"+c a' b" — b a' c"+b c' a" — c b' a'" 

Ex. 6. Given x (x + y + z) = 18^ 

y [x -f. y -f z) == 27 > to find x,y, and z. 
z (x + y + z) = 36 ) 

Jlns. x = 2, y = 3, z = 4. 

Ex. 7. Given (x + y) — = 60, and (x + y)— = 2|, to find 
y x 

x and y. *ftns. x = 10, y s= 2. 

jEor. 8. Given £z + |y-fi* = 62^ 

i^ + iy + y z = 4 7>to find x y y, and z. 

i^ + iy + i* = 38) 

^n«y. a? = 24, y = 60, * = 120. 



Solution of General Problems. 

A general algebraic problem is that in which all the quanti- 
ties concerned, both known and unknown, are expressed by let- 
ters, or other general characters. Not only such problems as 
have their conditions proposed in general terms are here im- 
plied ; every particular numerical problem maybe made general, 
by substituting letters for the known quantities concerned in it ; 
when this is done, the problem which was originally proposed in 
a particular form becomes general. 

In solving a problem algebraically some letter of the alphabet 
must be substituted for an unknown quantity. And if there be 
more unknown quantities than one, the second, third, &c. must 
either be expressed by means of their dependence upon the first 
and one or other of the data conjointly, or by so many distinct 
letters. Thus, so many separate equations will be obtained, the 
resolution of which, by some of the foregoing rules, will lead to 
the determination of the quantities required. 



ALGEBRA : GENERAL PROBLEMS. 85 



Examples. 

1. Given the sum of two magnitudes, and the difference of 
their squares, to find those magnitudes separately. 

Let the given sum be denoted by s, the difference of the 
squares by d ; and let the two magnitudes be represented by x 
and y respectively. 

Then, the first condition of the problem expressed algebraic- 
ally is x + y = s 

And the second is x 2 — y 2 = d 

Equa. 2 divided by equa. 1, gives x — y = — 

s 

D S 2 ID 

Equa. 1 added to equa. 3, gives 2 x = |- s = 

s s 



Equa. 4 divided by 2, gives x = 



Equa. 5 taken from equa. 1, gives y = s — 



2s 

s 2 + d 



2 s 2 s 

To apply this general solution to a particular example, sup- 
pose the sum to be 6, and the difference of the squares 12. 
Then s = 6 and d = 12, 

2s 12 12 

s 2 — b 36 — 12 24 n 

and y = = = — = 2. 

y 2 s 12 12 

Suppose, again, s = 5, d = 5 : 

25 + 5 a a 25 — 5 o 

then x = 1Q = 3, and y = — — = 2. 

Iftr. 2. Given the product of two numbers, and their quotient, 
to find the numbers. 

Let the given product be represented by p, the quotient by q\ 
and the required numbers by x and y, as before. 

Then we have, 1.xy=p 

x 
and 2. — = a. 

y 

Equa. 2 x by y, gives x = qy 
Substituting this value > 2 _ 
of x for it in equa. 15^^ ~~ -^ 

Dividing by q, y 2 = — 

13 I 



86 ALGEBRA : GENERAL PROBLEMS. 

Extracting the square root, y = I- 



Then, by substitution, x = q y = q \-= y—L. = 

s/pq. 
Suppose the product were 50 and the quotient 2. 

Then y = |£ = I— = >/ 25 = 5, and x = v' jo q = 

^100 = 10. 
Again, suppose the product 36, and the quotient 2 A. 

Then y == M = J— = </ 16 = 4, and x = %/ j» q — 

n/81 = 9. 

Ex. 3. Given the sum (s) of two numbers, and the sum of 
their squares s, to find those numbers. 

Jins. x = \ s + ± \/2 s — s 2 , and y = | s — | y/2 s — s\ 

Ex. 4. The sum and product of two numbers are equal, and 
if to either sum or product the sum of the squares be added, the 
result will be 12. What are the numbers ? 

Ans. each = 2. 

Ex. 5. The square of the greater of two numbers multi- 
plied into the less, produces 75 ; and the square of the less 
multiplied into the greater produces 45. What are the 
numbers ? 

Ex. 6. A man has six sons whose successive ages differ by- 
four years, and the eldest is thrice as old as the youngest. 
Required their several ages ?—J2ns. 10, 14, 18, 22, 26, and 30 
years. 



Section IX. — Quadratic Equations. 

When, after due reduction, equations assume the general 
form A# 2 + Ba? + c = 0; then dividing by a, the coefficient 

of the first term, there results x 2 -\ x -\ — = 0, or, making 

A A 

p — — , q = — , we have x 2 + p x -f q =0 (1) 

A A 

an equation which may represent all those of the second degree, 
p and q being known numbers positive or negative. 



ALGEBRA : QUADRATICS. 87 

Let a be a number or quantity which when substituted for x 
renders x 2 -\-p x+q=0; then a 2 -\- p a+q=0, or q— — a 2 — p a. 
Consequently x 2 -f p x + q, is the same thing as# 2 — a 2 -f p 
p x—p a, or as (x+a) (x — a) + p (x — a), or, lastly, as (x — a) 
(x + a+p). 

The inquiry, then, is reduced to this, viz. to find all the 
values of x which shall render the product of the above two 
factors equal to nothing. This will evidently be the case when 
either of the factors is = ; but in no other case. Hence, we 
have x — a,—0, and x+a+p=0, or x=a, and x = — a — p.* 

And hence we may conclude — 

1. That every equation of the second degree whose condi- 
tions are satisfied by one value a of x, admits also of another 
value — a — p. These values are called the roots of the quad- 
ratic equation. 

2. The sum of the two roots a and— a — p is — p; their pro- 
duct is — a 2 — a p, which as appears above is—q. So that the 
coefficient, p, of the second term is the sum of the roots with 
a contrary sign ; the knoivn term, q, is their product. 

3. It is easy to constitute a quadratic equation whose roots 
shall be any given quantities b and d. It is evidently x 2 — 
(b + d) x+b d=0. 

4. The determination of the roots of the proposed equation 
(1) is equivalent to the finding two numbers whose sum is — p, 
and product q. 

5. If the roots b and d are equal, then the factors x — b and 
x — d are equal ; and x 2 +p x-\-q is the square of one of them. 

To solve a quadratic equation of the form x 2 -\-p x-\rq=O f let 
it be considered that the square of x + -| p is a trinomial, x 2 + 
p x-\-\p 2 , of which the first two terms agree with the first two 
terms of the given equation, or with the first member of that 
equation when q is transposed. 

That is, with x*+p x= — q. 

Let then i p 2 be added, we have, x*+ px+% p % —\p 2 — q 
of which the first member is a complete square. 

* If it be affirmed that the given equation admits of another value of x, besides 
the above, b for instance, it may be proved as before that x — b must be of the num- 
ber of the factors of x 2 -\-p x-{-q, or of (x — a) (x-\-a-\- p). Butx — a and 
x-\-a^-p being prime to each other, or having no common factor, their product 
cannot have any other factor then they. Consequently b must either be equal to o 
or to — a — p ; and the number of roots is restricted to two. 



88 ALGEBRA : QUADRATICS. 

Its root is x 4- h p—± \/(\p* — q) 

and consequently x == — ^ jo ± </(\p* — q) 
otherwise, from number 2 above, we have x + x' = — jo 

and xx' = q 

Taking 4 times the second of these equations from the square 
of the first, there remains x 2 — 2 x x'-\-x' 2 =p 2 — 4 q 

Whence, by taking the root, x — x'= s/(p 2 — 4 q) 

Half this added to half equa. 1, gives a?= — h p-h 
^(/-4 ? ) = — hp+ s/(ip 2 —q) 

And the same taken from half equa. 1, gives x' — — h p — 
h \/{p 2 — 4 q) = — \ p — ^/(i p 2 — q) which two values of 
a evidently agree with the preceding. 

It would be easy to analyze the several cases which may 
arise, according to the different signs and different values, of p 
and q. But these need not here be traced. It is evident that 
whether there be given 

1. x 2 -\-p x—q 

2. x 2 — p x=q 

3. x 2 -\-p x— — q 

4. x* — p x— — q 

The general method of solution is by completing the square, 
that is, adding the square of \ p, to both members of the equa- 
tion, and then extracting the root. 

It may farther be observed that all equations may be solved 
as quadratics, by completing the square, in which there are 
two terms involving the unknown quantity or any function 
of it, and the index of one double that of the other. Thus, 

n n 

x 6 =b p x 3 = q, x sn ± p x n = q, x 2 ± p x 4 = q, (x* -f p x + q) 2 
=b (x 2 +p x+q)=r, (x 2n — x n ) 2 ± (x 2n — x n )=q, &c, are of the 
same form as quadratics, and admit of a like determination of 
the unknown quantity. Many equations, also, in which more 
than one unknown quantity are involved, may be reduced to 
lower dimensions, by completing the square and reducing ; such, 

for example, as (x*+y z ) z ±/?(z 3 +y 3 ) = q,— j±^— = q, and 

y " 
so on. 

Note. — In some cases a quadratic equation may be conve- 
niently solved without dividing by the coefficients of the 
square, and thus without introducing fractions. To solve the 
general equation a x 2 +_ b x — c, for example, multiply the 
whole by 4 a, whence 4a*x 2 +_4abx = 4ac, adding b 2 to 
complete the square, 4 a 2 x 2 ±_ 4 a b x+b 2 =4 a c+6 9 
taking the square root, 2 a x + b = + ^/(4 a c+b 2 ) ; 



ALGEBRA I QUADRATICS. 89 

whence x = : which will serve tor a 

2 a 

general theorem. 



Examples. 

1. Given a? 3 — 8 x + 10 = 19, to find a\ 

transposing the 10, x 2 — 8 x = 19 — 10=9 
completing the squ., x 2 — 8 x+ 16 = 9 + 16=25 
extracting the root, x — 4= ± 5 
consequently ic=4 ± 5=9 or — 1. 

10 14 — 2x 22 

2. Given = =—, to find the values of x. 

x x* 9 

22 x B 
multiplying by a? 2 , 10 a? — 14 + 2 x = — - — , 

y 

transposing, 2 T 2 x* — 12 x= — 14, 

dividing by % 2 , z z — \± x = — f *, 

complet. squ. *»— f< x+ ft X) 3 = fff — f > = T V T , 

extract, root, a: — fr " =*= ttj 

transposing, a? = f | ± T 6 T = 3 or fl. 



3. Given z 3 +2 a?+4 x/a: 2 +2 a?+l=44, to find x. 

adding 1, we have (x 2 + 2 a?+l) + 4 \/(a? 3 + 2 x + 1) 
45 

complet squ. (x* + 2 a? + 1) + 4 s/(x*+2 x + l) + 4 
49 

extract, root, x/(a? 3 +2 a?+l) + 2 = ± 7 

transposing the 2 ^(ar 3 + 2 a? + 1) = ± 7 — 2 = 5 oi 
-9 

that is, x + 1 = 5 or — 9 

hence x = 4 or — 10. 



4. Given a? n — 2 « a? 3 = c, to find x 

n 

complet. squ. # n — 2 a x* + a 3 =c+a 3 

n 

extract, root, x* — a = ± >/(c+« 2 ) 
transposing, a? 2 = a + \/(c+a 2 ) 



consequently, a: = (a + v'c+a 8 ) 1 
J 2 



90 ALGEBRA : QUADRATICS. 

a? 2 4 a? 

5. Given — -\ = 12, and x — y = 2, to find x and y. 

X* x 
complet. squ. in equa. 1, — +4- +4=16. 
«y if 

x x 

Extracting root — -f 2 = ± 4 : whence — = 2 or — 6, and 
«y if 

x=2 y or = — 6 y. 

Substituting the former value of x in the 2d equa. it becomes 
2 y — y—2, or y=2 ; whence a?=4. 

Again, substituting the 2d value of x, in equa. 2, it becomes 
— 6 2/ — V or — 7y = 2; whence y = — f , and a? = + V 2 - 

6. Given a? 2 y* — 5=4 x y, and § a? y = f y 3 , to find 
a? and y. 

equa. 1, by transposition, becomes x* y 2 — 4 a? y—5 
completing the square, x 3 y 2 — 4 a? y-j-4=9 
extracting the root, a? y — 2 = ±3 
Whence a? y=5 or — 1. 

Substituting the first of these values for a? y in equa. 2, it be- 
comes J- # 3 =f : whence y=l and a?=5. 

Substituting the 2d value in the same equation, it becomes 
f y 3 = — § : whence y = — -^| = — i \^25, and a? = — 1 
-T-— 1^25= ^5. 

a? 3 6 351 

7. Given ,— —4- — =-^- — r, to find a?. 

(a: 3 — 4) 3 ^a? 3 — 4 25 a? 2 ' 

•tfrcs. a? = ± 3, or ± y/%*. 

8. A man travelled 105 miles at a uniform rate, and then 
found that if he had not travelled so fast by two miles an hour, 
he would have been six hours longer in performing the same 
journey. How many miles did he travel per hour ? 

Jlns. 7 miles per hour. 

9. Find two such numbers that the sum, product, and dif- 
ference of their squares may be equal. 

Jlns. i + i ^/5, and | + i ^/5. 

1 0. A waterman who can row eleven miles an hour with the 
tide, and two miles an hour against it, rows five miles up a river 
and back again in three hours : now, supposing the tide to run 
uniformly the same way during these three hours, it is required 
to find its velocity ? 

Arts. 4 if miles per hour. 



algebra: equations. 91 



Section X. — Equations in General. 



Equations in general may be prepared or constituted by the 
multiplication of factors, as we have shown in quadratics. Thus, 
suppose the values of the unknown quantity x in any equation 
were to be expressed by «, b, c, d, &c. that is, let x = a, x — b, 
x — c,x=d, &c. disjunctively, then will x — a—0, x — 6=0, 
x — c = 0, x — d = 0, &c. be the simple radical equations 
of which those of the higher orders are composed. Then, as the 
product of any two of these gives a quadratic equation ; so the 
product of any three of them, as (x — a) (x — b) (x — c) = 0, 
will give a cubic equation, or one of three dimensions. And 
the product of four of them will constitute a biquadratic equa- 
tion, or one of four dimensions ; and so on. Therefore, in ge- 
neral, the highest dimension of the unknown quantity x is 
equal to the number of simple equations that are multiplied 
together to produce it. 

When any equation equivalent to this biquadratic (x — a) 
[x — b) (x — c) (x — d) = is proposed to be resolved, the 
whole difficulty consists in finding the simple equations x — a 
= 0, x — b = 0, x — c = 0, x — d =0, by whose multiplica- 
tion it is produced ; for each of these simple equations gives one 
of the values of x, and one solution of the proposed equation. 
For, if any of the values of x deduced from those simple equa- 
tions be substituted in the proposed equation in place of x, then 
all the terms of that equation will vanish, and the whole be 
found equal to nothing. Because when it is supposed that x — a, 
or x = b, or x = c, or x = d, then the product (x — a) (x — b) 
(x — c) (x — d) vanishes, because one of the factors is equal to 
nothing. There are, therefore, four suppositions that give 
(x — a) (x — b) (x — c) (x — d) = 0, according to the pro- 
posed equation ; that is, there are four roots of the pro- 
posed equation. And after the same manner any other equa- 
tion admits of as many solutions as there are simple equa- 
tions multiplied by one another that produce it, or as many as 
there are units in the highest dimensions of the unknown quan- 
tity in the proposed equation. 

But as there are no other quantities whatsoever besides 
these four (a, b, c, d,) that, substituted in the proposed product 
in the place of x, will make that product vanish : therefore, 
the equation (x — a) (x — b) (x — c) (x — d) — 0, cannot 
possibly have more than these four roots, and cannot admit 



92 ALGEBRA I EQUATIONS. 

of more solutions than four. If we substitute in that product 
a quantity neither equal to 0, nor b, nor c, nor d, which suppose 
e, then since neither, e — a,e — b, e — c, nor e — d, is equal to 
nothing ; their product cannot be equal to nothing, but must be 
some real product : and therefore, there is no supposition be- 
sides one of the aforesaid four, that gives a just value of x ac- 
cording to the proposed equation. So that it can have no more 
than these four roots. And after the same manner it appears, 
that no equation can have more roots than it contains dimen- 
sions of the unknown quantity. 

To make all this still plainer by an example in numbers, 
suppose the equation to be resolved to be x 4 — 10 x 3 -f 35 x 2 — 
50 x + 24 = 0, and that we discover that this equation is the 
same with the product of (a? — 1) (x — 2) (x — 3) (x — 4), 
then we certainly infer that the four values of x are 1, 2, 3, 4 ; 
seeing any of these numbers, placed for x, makes that product, 
and consequently x 4 — 10 x 3 -f- 35 # 3 — 50 x + 24, equal to 
nothing, according to the proposed equation. And it is certain 
that there can be no other values of x besides these four ; for 
when we substitute any other number for x in those factors 
x — 1, x — 2, x — 3, x — 4, none of them vanish, and therefore 
their product cannot be equal to nothing, according to the equa- 
tion. 

A variety of rules, some of them very ingenious, for the so- 
lution of equations, may be found in the best writers on Alge- 
bra ;* but we shall simply exhibit the easy rule of Trial-and- 
Error, as it is given by Dr. Hutton in the 1st volume of his 
" Course of Mathematics." 



Rule. 



" 1. Find, by trial, two numbers as near the true root as pos- 
sible, and substitute them in the given equation instead of the 
unknown quantity ; marking the errors which arise from each 
of them. 

" 2. Multiply the difference of the two numbers, found by 
trial, by the least error, and divide the product by the differ- 
ence of the errors, when they are alike, but by their sum when 
they are unlike. Or say, as the difference or sum of the errors 
is to the difference of the two numbers, so is the least error to 
the correction of its supposed number. 

* See the treatises of Lacroix, Bonnycastle, Wood, J. R. Young, &c. 



ALGEBRA : EQUATIONS. 



93 



" 3. Add the quotient, last found, to the number belonging to 
the least error, when that number is too little, but subtract it 
when too great, and the result will give the true root nearly. 

" 4. Take this root and the nearest of the two former, or any- 
other that may be found nearer ; and, by proceeding in like man- 
ner, a root will be had still nearer than before • and so on to any 
degree of exactness required. 

" Note. — It is best to employ always two assumed numbers 
that shall differ from each other only by unity in the last figure 
on the right ; because then the difference, or multiplier, is 1." 



Example. 



To find the root of the cubic equation x* + x* -f x — 100, or 
the value of x in it. 



Here it is soon found that x 
lies between 4 and 5. Assume, 
therefore, these two numbers, 
and the operation will be as fol- 
lows: 



Again, suppose 4*2, and 4*3, 
and repeat the work as follows: 



the sum of which is 71. 
Then as 71 : 1 :: 16 : '225. 
Hence x = 4*225 nearly. 



1st Sup. 
4 . . . 
16 . . . 
64 . . . 


2d Sup. 
. x .... 5 
. x 2 . . . . 25 
. x s .... 125 


1st Sup. 

4-2 
17-64 
74-088 


. . X 

. . X 2 . 
. . X 3 . 

. . sums . 
. . errors 


2d Sup. 
4-3 
. 18-49 
. 79-507 


84 . . . 


sums . . . 155 


95-928 


102-297 


—16 . . . 


errors . . . -\-55 


— 4-072 


. +2-297 



the sum of which is 6-369. 
As 6-369 : -1 : : 2-297 : 0-036 
This taken from . . 4-300 



leaves x nearly = 4-264 



14 



94 ALGEBRA : EQUATIONS. 

Again, suppose 4-264 and 4*265, and work as follows : 

4-264 .... x 4-265 

18-181696 .... x* 18-190225 

77-526752 .... X s 77*581310 



99-972448 



sums 



100-036535 



027552 



errors 



+ 0-036535 



the sum of which is -064087 
Then as -064087 : -001 :: -027552 : 0-0004299 
To this adding . . . 4-264 



gives a? very nearly = 4*2644299 

When one of the roots of an equation has been thus found, 
then take for a dividend the given equation with the known 
term transposed to the unknown side of the equation ; and for 
a divisor take x minus the root just determined : the quotient 
will be equal to nothing, and will be a new equation de- 
pressed a degree lower than the former. From this a new 
value of x may be found : and so on, till the equation is re- 
duced to a quadratic, of which the roots may be found by the 
proper rules. 



Section XI. — Progression. 

When a series of terms proceed according to an assignable 
order, either from less to greater or from greater to less, by con- 
tinual equal differences or by successive equal products or quo- 
tients, they are said to form & progression. 

If the quantities proceed by successive equi-diflferences they 
are said to be in Arithmetical Progression. But if they pro- 
ceed in the same continued proportion, or by equal multiplica- 
tions or divisions, they are said to be in Geometrical Pro- 
gression. 

If the terms of a progression successively increase, it is called 
an ascending progression : if they successively decrease, it is 
called a descending progression. 

Thus, 1, 3, 5, 7, 9, &c. form an ascending arithmetical 
24, 22, 20, 18, 16, &c. form a descending arithmetical 
1, 3, 9, 27, 81, &c. form an ascending geometrical 
4, 2, 1, \, i, &c. form a descending geometrical - 



r 2 



ALGEBRA • PROGRESSION. 95 



Arithmetical Progression. 

1. Let a be the first term of an arithmetical progression 

d the common difference of the terms 
z the last term 
n the number of terms 
s the sum of all the terms. 
Then a, a+d, a-\-2 d, a + 3 d, &c. is an ascending progression, 
and a, a — d, a — 2 d,a — 3d, &c. a descending progression. 
Hence, in an ascending progression, a + (n — 1) d, is the last 

term ; 
in a descending progression, a — (n — 1) d, is the last 

term . 

2. Let a series be a -f (« + d) + (a -f 2 d) + (a+3 c?). 
The same inverted, (a + 3 d) + (a + 2 d) + (a + d) + a. 

The sum of the two, (2 a + 3 d) + (2 a -f 3 tf) + (2 a + 3 a?) 
+ (2 a + 3 d ) = 2 s. 

That is, (2 a -f 3 d ) x 4, in this case (« -f a -f 3 d ) n=2 s. 

Consequently, s = h {a + a = 3 d) n, or=i (a + 2) n, since 
2 is here=a -{-3d. The same would be obtained if the pro- 
gression were descending ; and let the number of terms be what 
it may. 

3. From the equations z=a + (n — 1) d, s = h. n {a -f z), 
and s=3 n (a-\-a+(n — 1) d), we may readily deduce the fol- 
lowing theorems applicable to ascending series. When the se- 
ries is descending, either the signs of the terms affected with d 
must be changed, or a must be taken for z ; and vice versa. 

(1.) a = z — nd + d= s /(—2sd +z 2 + dz + id 2 ) + 

\ d = — 1-3 d — \n d= z. 

n n 

ln . j z — a z* — a z 2zn — 2s 2s—2na 

(2.) G? = = = = — 

n — 1 2 s — z — a n* — n n* — n • 

a 

(3.) z = a+nd — d=-+hnd — id — 
n 

</(2sd+ a* — ad + id 2 )—% d = — —a. 

n 

(4.) *=4n («+*)=(«+£ nd—hd) n={z—hnd+hd) n 

z* — a* + d{z + a) 
~ 2d 

(5.) n = — ; — — -r— 1. 

' a + z d 



96 ALGEBRA : PROGRESSION. 



Examples. 

1. Required the sum of 20 terms of the progression, 1, 3, 5, 

7, 9, &c. 

Here a = 1, d =2, n = 20 ; which is substituted in the 
theorem s = (a + h n d — \ d) n, transform it to s = 
(1 + 20 — 1) 20 == 20 X20 = 400, the sum required. 

Note. — In any other case the sum of a series of odd numbers 
beginning with unity, would be = n 3 , the square of the number 
of terms. 

2. The first term of an arithmetical progression is 5, the last 
term 41, the sum 299. Required the number of the terms, and 
the common difference. 

2 s 598 

Here n = = — — = 13, the number of terms, 

a +z 46 

41 5 

__ — -- — = 3, the common difference. 
n — 1 12 

There are 8 equidifferent numbers : the least is 4, the greatest 
32. What are the numbers ? 

Here d = — = — - — ■= 4, the common difference. 

n — 1 7 

Whence 4, 8, 12, 16, 20, 24, 28 9 32, are the numbers. 

4. The first term of an arithmetical progression is 3, the num- 
ber of terms 50, the sum of the progression 2600. Required 
the last term and the common difference. 

2 s 5200 
Here z = a =— 3 = 104 — 3 = 101, 

n 50 ■ 

the last term, 

, , 2 — a 101 — 3 n . • 

and d = — = — — = 2, the common difference. 

n — 1 49 

5. The sum of six numbers in arithmetical progression is 48 ; 
and if the common difference d be multiplied into the less ex- 
treme, the product equals the number of terms. Required those 
terms, Ans. 3, 5, 7, 9, 11, and 13. 



algebra: progression. 97 



Geometrical Progression. 

1. Let a be the first term of a geometrical series; 

r the common ratio; z the last term; 

n the number; and s the sum of the terms. 
Then a, r a, r 2 a, r* a, r 11 - 1 a, is a geometrical progression, 
which will be ascending or descending, according as r is an in- 
teger or a fraction. 

2. Let the prog, a+r a+r 2 a+r* a+r 4 a=s, be x by r, it 
becomes r a+r 2 a+r 3 a+r 4 a+r 6 a=r s 

The diff. of these is, — a + r 5 a =r s — s. 

But r s a is the last term of the original progression multi- 
plied by r, or in general terms r n ~ l axr, that is, r n a. Conse- 
quently r n a — a = r s — s 

„_, r n a — a r n — 1 ., c , , 

Whence s = = a 9 the sum of the series. 

r — 1 r — 1 

A similar method will lead to a like expression for s, what- 
ever be the value of n. If r be a fraction, the expression 

1 r n 

becomes transformed to s =; a. 

1 — r 

3. Now from these values of z and s the following theorems 
may be deduced. 

. . z (r — l)s 

(1.) a = -— - 1 =- L - 7- =s+r z — r s. 

rs — s+a s {r n — r n ~ l ) 

(2.) z=ar n ~ 1 =— =— — — '. 

* * r r* — • 1 



. _ a(r n — \)_a(l — r n )_ r z — a _ 

I 1 •' S ~~ r — l~~ l—r r— 1 



zr ft — z 



y.™ ___ 1** 



(4.) r = 1-)*" 1 . 

' s — z \af 

And, if the logari 
that of r=R : then, 



And, if the logarithm of-= n, that of = m, and 

K 



98 ALGEBRA : PROGRESSION. 



(5.) „ = ?+! = *. 

V ' R R 



Also, if when r is a fraction, n is infinite, then is r n =0, and 
the expression for s becomes 

(6.) s = [This expression is often of use in the sum- 
mation of infinite series.] 

Examples. 

1. The least of ten terms in geometrical progression is 1, the 
ratio 2. Required the greatest term, and the sum. 

Here z — a r n ~ 1 = 1 x 2 9 =512, the greatest term ; 

r z — a 1022 — 1 

and s = -— = ; =1023, the sum. 

r — 1 1 

2. Find the sum of 12 terms of the progression 1, -i, £, Jfo &c. 

1 _ r n 1 — -^tVtt 265720 ' 

Here s=- = , 531 1 441 = 1 ^ 1 ^ , the sum. 

1 — r 1 — 4- 177147 



3. Find the sum of the series 1, h, i, s, &c. carried to infinity. 

, , becomes s = - — , 

I — r l — | 

the sum required. 



Here theor. 6, that is s = — — - , becomes s = = 2, 



4. Find the vulgar fraction equivalent to the circulating de- 
cimal -36363636. 

This decimal, expressed in the form of a series, is, T 3 ^j--f 
To 3 oW+ T oo 3 oW+ &c - where «=T<n>> and r = T hr 

Consequently, s = = T 3 /o ■+■ tVo ** tt> the fraction 

sought. 

5. Find the sum of the descending infinite series 1 — x + 
x* — aP+x 4 , &c. 

Here a= 1, r— — x, and s=- =- — ■ — , the sum req. 

1 — r 1 + z ^ 

And, by way of proof, it will be found that if 1 be divided 

by 1 -\-x, the quotient will be the above series. 

6. Of four numbers in geometrical progression the product 
of the two least is 8, and of the two greatest 128. What are the 
numbers? Ans. 2,4, 8, and 16. 



ALGEBRA : LOGARITHMS. 99 



Section XII. — Logarithms. 



The logarithm of any given number is the index of such a 
power of some other number, as is equal to the given one. 

Let us suppose that the number r greater than unity, is the 
base of a system of logarithms, and let there be given to it the 
variable exponent^, in such manner that the expression r* shall 
represent all possible numbers, by attributing successively dif- 
ferent values to the exponent p. It is manifest, 

1. That the logarithm of unity will be always zero or nothing, 
whatever be the base r : for, in general, r°=\ 

2. That the logarithm of the base r will be 1, since r is the 
same thing as r\ 

3. That all numbers above 1 will have positive numbers for 
their logarithms. Thus, supposing r= 10, then the number 
10000 or 10 4 has for its logarithm the positive number 4. 

4. All fractions, or numbers below unity, have negative num- 
bers for their logarithms. Thus, if r=10, then T q ±^-$ or -0001 
or 10 -4 has — 4 for its logarithm. 

5. Assuming two numbers n and n', to which correspond 
respectively the two logarithms p and p', to the same base or 
root r : we have N=r p , and N'=r p ', and consequently N-f n' 
—r p xr p '—r p+v '. Whence it appears that in every system of 
logarithms, the logarithm p+p' of a product n n', composed of 
two factors, is equal to the sum of the logarithms of those fac- 
tors. 

6. If we have any numbers a, b, c, d, how many soever, 
we may prove in a similar manner, that (using the initial A to 
denote the logarithm) we shall have A(a.b.c.d) = ^a4- 
X b+A c+A d. 

7. If a = b = c = d, we shall have A (a X A X A X a), or A a 4 
= A a+A a+A a+A a = 4 A a : and in general A A n = n A a. 
Thus it appears that the logarithm of any integral positive 
power, n, of any number a, is equal to n times the logarithm 
of A. 

— n 

8. We have also A a* = — A, a (n and p being positive in- 

n n 

tegers). For, let a 7 = k, and consequently A a f = A k. From 

n 

the equation a* = x, we have by raising the whole to the 
power p, A n = k v , and, of consequence, n A a«= p X x, or by 

n . z. 

division -A a = A x = A a p . 



100 algebra: logarithms. 



9. From the same principle it follows, that ^- = Aa — A b, 

B 

For, let— =q, and consequently a=b X Q : we shall then have 

B 

A a=A b+A Q ; whence A Q=A a — A b. So that the logarithm 
of the quotient is equal to the logarithm of the dividend minus 
that of the divisor ; or the logarithm of a fraction is equal to 
the logarithm of the numerator made less by that of the deno- 
minator. 

1 

10. Farther, A, A~ n = — n A a. For A -n = — : therefore 

A A~ n = A1 — AA n = — wAa: which is no other than 
— n A a. 

-- n -- - 

11. Again, A a * s= A A. For a v = 1~a*: whence 

— — n n 

results A a v — A a= A a. 

V P 

12. Suppose there be two systems of logarithms whose roots 
or bases are r and s. Let any number n have p for its loga- 
rithm in the first system, and q for its logarithm in the second : 

we shall have N=r v and N—s q ; which gives r v =s q , and s = r q . 
Therefore, taking the logarithms for the system r, we shall have 

As = — Ar; or, if in the system rwe have A r—l, then A *=— , 

p 1 

or g = - — = p x r — . Thus, knowing the logarithm p of any 

A<s A s 

number n, for the system whose base is r, we may obtain the 
logarithm q of the same number for the system s, by multiply- 
ing p by a fraction whose numerator is unity and denominator 
the logarithm of s taken in the system r. 

1 3. In the system of logarithms first constructed by Baron 
Napier, the great inventor, r = 2-718281828459, &c. and the 
exponents are usually denominated Napierian or Hyperbolic 
logarithms ; the latter name being given because of the 
relation between these logarithms and the lines and asymptotic 
spaces in the equilateral hyperbola : so that in this system n 
is always the hyperbolic logarithm of (2-71828, &c.) n . But 
in the system constructed by Mr. Briggs (corresponding with 
the spaces in a hyperbola whose asymptotes make an angle 
of 25° 44' 25" 28'"), called common or Briggean logarithms, 
r — 10 ; so that the common logarithm of any number is the 
index of that power of 10 which is equal to the said number. 



ALGEBRA : LOGARITHMS. 101 

•69897 3*95535 1 

Thus, if 50 = 10 , and 9023 = 10 ; then is 1*69897 

the common logarithm of 50, and 3*955351 the common loga- 
rithm of 9023. 

14. The rules for the management and application of loga- 
rithms being given in the best collections of logarithmic tables, 
are here omitted. The tables published in England by Dr. Hut- 
ton, those published in France by Callet, and those recently 
published by Professor Babbage, may be recommended as the 
most correct and best fitted for scientific use. Mr. GalbraiWs 
tables are correct and valuable. 

If the reader wish for neatly arranged tables in small com- 
pass, for the practical purposes of a man of business, I would 
recommend those of Mr. Woollgar, given in the Mechanics' 
Magazine, those recently published by Mr. J. R. Young, and 
those given by Mr. Carr, in his valuable " Synopsis of Prac- 
tical Philosophy." 



Section XIII. — Computation of Formulae. 

Since the comprehension, and the numerical computation of 
formulae expressed algebraically, are of the utmost consequence 
to practical men, enabling them to avail themselves advanta- 
geously of the theoretical results of men of science, as well as 
to express in scientific language the results of their own experi- 
mental or other researches ; it has appeared expedient to present 
brief treatises of Arithmetic and Algebra. The thorough un- 
derstanding of these two initiatory departments of science will 
serve essentially in the application of all that follows in the pre- 
sent volume ; and that application may probably be facilitated 
by a few examples, as below : — 

Ex. 1. Leta=5, 6=12, c=13, and s=a+b-\-c ; then what 
is the numerical value of the expression, 

\/[h s {h s — a) (h s — b) (h s — c)], which denotes the area 
of the triangle whose sides are 5, 12, and 13 ? 

Heres=a+6+c=5 + 12 + 13=30 . ± 5== i5 ; 
i s — «=15 — 5=10 ; h 5 — 6=15 — 12=3 ; h s — c 
= 15 — 13=2. 

Consequently, by substituting the numerical values of the 
several quantities between the parentheses for them, we shall 
have 

^(15 X 10 X 3 x 2)=v/900=30, the value required. 

Ex. 2. Suppose g = 32J, / = 6 : required the value of 
h g t 2 , an expression denoting the space in feet which a heavy 
15 k2 



102 NUMERICAL COMPUTATION OF FORMUL2E. 

body would fall vertically from quiescence in six seconds, in 
the latitude of London. 

Here § g * 2 =16 T V X 6 2 =96£ X 6 = 579 feet. 

Ex. 3. Given d=6, rf=4, A = 12, * - 3-141593 ; required 
the value of A * A (d'+d tf+rf 3 ), a theorem for the solid content 
of a conic frustrum whose diameters of the two ends are d, d 9 
and height h. 

Here n'=36, » <Z=6 X 4=24, «P = 16,^ = '2618 nearly. 

Hence -i- * A (d 3 +d d+rf«) = -2618 (36+24+16) 12 
="e?8 X76 X 12=?141593 X 76=238-761068. 

Fa; 4 Let a=l,A=25, 2=193 inches: what is the value 
of 2 a ,/«■ h ? This being the expression for the cubic inches 
of water discharged in a second, from an orifice whose area i » a, 
and depth below the upper surface of water in the vessel or re- 
servoir, h, both in inches. q. 8 q 2 44 

Here 2 a •* A=2 ^(25 X 193)=10 ^193 = 10 X 13 89244 

— 138*9244 cubic inches. _ , 

Mr 5 Suppose the velocity of the wind to be known m 
miferperhour; required short approximative expressions for 
the yards per minute, and for the feet per second. 
First 1760H-60=V=29i=30 nearly. 

Also 5280^(60 xSoJ-ttWj-K-H- 14 "w- 
If, therefore, n denote the number of miles per hour 

30 n will express the yards per minute ; and U n, the teet 

Per The C s°e nd are approximative results : to render them correct 
where complete accuracy is required, subtract from each result 
its 45th part, or ite fifth part of its ninth part. 

Thus suppose the wind blows at the rate of 20 miles per 

h ° Ur Then 30 »=30 X 20 = 600 yards per minute, or more cor- 
rectly 600 - W= 600 - 134=586| yards. 
Also U »=30 feet per second, 
or correctlv, 30 — M=30— |=29§feet. 
Converse V fk the feet per second will indicate the m.les 
per hour, co/rect within the 45th part, which is to be added to 
obtain the true result. , 

&r 6 To find a theorem by means of which it may be as 

certained when a general law exists, and 7 ,h f.^Z^ m the 

Sunnose for example, it were required to determine tne 

law which'prevailed between the resistances of bodies moving 

„ the air a'nd other resisting media, and *e velocities w * 

which they move. Let v, v, denote any two velocities, and B,r 



COMPUTATION OF FORMULA. 103 

the corresponding resistances experienced by a body moving 
with those velocities : we wish to ascertain what power of v it 
is to which r is proportional. Let x denote the index or ex- 
ponent of the power ; then will v 1 : v x : : r : r, if a law subsist. 

/ V\ x T 

Div.the consequents by the antecedents, we have 1 : ( - ) : : 1 : -. 

\ v/ R 

/ V\ x T 

Consequently (-) =-. This, expressed logarithmically, 

v v 

gives x X log. - = log.- ; 

log. r — log;. R 

or x = —2 _o — . 

log. v — log. v 

Hence, the quotient of the differences of the logs, of the re- 
sistances, divided by the difference of the corresponding velo- 
cities, will express the exponent x required. 

This theorem is of very frequent application in reference to 
the motion of cannon balls, of barges on canals, of carriages on 
rail roads, &c. and may indeed be applied to the planetary 
motions. 



104 PLANE GEOMETRY. 



CHAPTER III. 
PRINCIPLES OF GEOMETRY. 

Definitions. 

1. Geometry is a department of science, by means of which 
we demonstrate the properties, affections, and measures of all 
sorts of magnitude. 

2. Magnitude is a continued quantity, or any thing that is 
extended ; as a line, surface, or solid. 

3. A point is that which has no parts : i. e. neither length, 
breadth, nor thickness. 

4. A line is a length without breadth or thickness. 
Cor. The extremes of a line are points. 

5. A right line is that which lies evenly, or in the same di- 
rection, between two points. A curve line continually changes 
its direction. 

Cor. Hence there can only be one species of right lines, but 
there is infinite variety in the species of curves. 

6. An angle is the inclination of two lines to 
one another, meeting in a point, called the an- 
gular point. When it is formed by two right 
lines, it is a plane angle, as a ; if by curve lines, 
it is a curvilinear angle. 

7. A right angle is that which is made by 
one right line a b falling upon another c d, 
and making the angles on each side equal, 
abc = abd; so that a b does not incline 
more to one side than another : a b is called a £_ 
perpendicular. All other angles are called 
oblique angles. 

8. An obtuse angle is greater than a right 
angle, as r. 




9. An acute angle is less than a right angle, as s. 

10. Contiguous angles are those made by one line falling 
upon another, and joining to one another, as r, s. 



PLANE GEOMETRY. 105 



11. Vertical or opposite angles, are those made 
on contrary sides of two lines intersecting one 
another, as a, b. 



12. A surface has only length and breadth. The extremes 
or limits of a surface are lines. 

13. A plane is that surface which lies perfectly even be- 
tween its extremes ; or in which, right lines any way drawn 
coincide. 

14. A solid is a magnitude extended every way, or which 
has length, breadth, and depth. 

The terms or extremes of a solid are surfaces. 

15. The square of a right line is the space included by four 
right lines equal to it, set perpendicular to one another. 

1 6. The rectangle of two lines is the space included by four 
lines equal to them, set perpendicular to one another, the oppo- 
site ones being equal. 




Sec. I. — Of Angles and Right Lines, and their Rectangles. 

Prop. 1. If to any point c in a right line a b, 
several other right lines d c, e c are drawn on 
the same side ; all the angles formed at the 
point c, taken together, are equal to two right 
angles, acd + dce +ecb = two right 
angles. 

Cor. 1. All the angles made about one point in a plane, be- 
ing taken together, are equal to four right angles. 

Cor. 2. If all the angles at c, on one side of the line a b, 
are found to be equal to two right angles : then A c b is a 
straight line. 

2. If two right lines, a b, c d, cut one 
another, the opposite angles e and g will be 
equal. 

3. A right line, b h, which is perpendicular to one of two 
parallels, is perpendicular to the other. 

4. If a right line c g, intersects two parallels _AJJ 
a d, f h ; the alternate angles, abe, and beh 
will be equal. 




F~7~ 



106 PLANE GEOMETRY I ANGLES, RECTANGLES, &C. 

Cor. 1. The external angle c b d, is equal to the internal 
angle on the same side b e h. 

Cor. 2. The two internal angles on the same side are equal 
to two right angles dbe + beh = two right angles. 

5. Right lines parallel to the same right line, are parallel 
to one another. 

g hi 

6. If a right line a c be divided into two 
parts, a b, b c : the square of the whole line is 
equal to the squares of both the parts, and 
twice the rectangle of the parts, a c 3 = A B a + 

B C 3 + 2 A B . B C. 

A B 

7. The square of the difference of two lines a c, b c, is equal 
to the sum of their squares, wanting twice their rectangle, 

A B a = A C 3 + B C 3 ' — 2 A C . B C. 

8. The rectangle of the sum and difference of two lines is 
equal to the difference of their squares. 

9. The square of the sum, together with the square of the 
difference of two lines, is equal to twice the sum of their 
squares. 



D E 



Section II. — Of Triangles. 
Definitions. 

1. A triangle is a plane figure bounded by three right lines, 
called the sides of the triangle. 

2. An equilateral triangle is one which has three equal 
sides. 

3. An equiangular triangle is one which has three equal 
angles ; and two triangles are said to be equiangular, when 
the angles in the one are respectively equal to those in the 
other. 

4. An isosceles triangle has two sides equal. 

5. A right-angled triangle is that which has a right angle. 
The side opposite to the right angle is called the hypothenuse. 

6. An oblique triangle is one having oblique angles. 

7. An obtuse angled triangle has one obtuse angle. 

8. An acute angled triangle has three acute angles. 

9. A scalene triangle has three unequal sides. 

10. Similar triangles are those whose angles are respective- 
ly equal, each to each. And homologous sides are those lying 
between equal angles. 

1. The base of a triangle, is the side on which a perpen- 




PLANE GEOMETRY ', TRIANGLES. 107 

dicular is drawn from the opposite angle called the vertex ; 
the two sides, proceeding from the vertex, are called the legs. 

Prop. 1. In any triangle a b c, if one side 
b c be produced or drawn out ; the external an- 
gle a c d will be equal to the two internal oppo- 
site angles a b. 

c D 

2. In any triangle, the sum of the three angles is equal to two 
right angles. 

Cor. 1. If two angles in one triangle be equal to two angles 
in another : the third will also be equal to the third. 

Cor. 2. If one angle of a triangle be a right angle, the sum 
of the other two will be equal to a right angle. 

3. The angles at the base of an isosceles triangle, are 
equal. 

Cor. 1. An equilateral triangle is also equiangular ; and the 
contrary. 

Cor. 2. The line which is perpendicular to the base of an 
isosceles triangle, bisects it and the verticle angle. 

4. In any triangle, the greatest side is opposite to the greatest 
angle, and the least to the least. 

5. In any triangle a b c, the sum of any 
two sides b a, a c, is greater than the 
third b c, and their difference is less than 
the third side. 




6. If two triangles a b c, a b c, have 
two sides, and the included angle equal in 
each ; these triangles, and their corres- 
ponding parts, shall be equal. 

7. If two triangles a b c and a b c, have two angles and an 
included side equal, each to each ; the remaining parts shall 
be equal, and the whole triangles equal. 

8. If two triangles have all their sides respectively equal : 
all the angles will be equal, and the wholes equal. 

9. Triangles of equal bases and heights are equal. 

10. Triangles of the same height, are in proportion to one 
another as their bases. 

11. If a line d e be drawn parallel to one side 
B c, of a triangle ; the segments of the other 
sides will be proportional ;ad:db::ae:ec. 

c 
Cor. 1. If the segments be proportional, ad:db::ae:ec; 

then the line d e is parallel to the side b c. 

Cor. 2. If several lines be drawn parallel to one side of a 

triangle, all the segments will be proportional. 




108 PLANE GEOMETRY I TRIANGLES. 

Cor. 3. A line drawn parallel to any side of a triangle, cuts 
off a triangle similar to the whole. 





12. In similar triangles the ho- 
mologous sides are proportional ; 

a b : a c : : d e : d f. 

c 

1 3. Like triangles are in the duplicate ratio, or as the squares 
of their homologous side. 

14. In a right angled triangle bac, 
if a perpendicular be let fall from the 
right angle upon the hypothenuse, it will 
divide it into two triangles similar to one 
another and to the whole, abd,adc. b o d 

Cor. 1. The rectangle of the hypothenuse and either seg- 
ment is equal to the square of the adjoining side. 

15. The distance a o of the right angle, from the middle 
of the hypothenuse is equal to half the hypothenuse. 

16. In a right-angled triangle, the square of the hypothenuse 
is equal to the sum of the squares of the two sides. 

17. If the square of one side of a triangle be equal to the sum 
of the squares of the other two sides ; then the angle compre- 
hended by them is a right angle. 

18. If an angle a, of a triangle b a c, be bi- k . 
sected by a right line a d, which cuts the a ...-•"' / 
base ; the segments of the base will be propor- 
tional to the adjoining sides of the triangle ; 
bd:dc::ab :ac. b d c 

19. If the sides be as the segments of the base, the line a d 
bisects the angle a. 

20. Three lines drawn from the three angles of a triangle to 
the middle of the opposite sides, all meet in one point. 

21. Three perpendicular lines erected on the middle of the 
three sides of any triangle, all meet in one point. 

22. The point of intersection of the three perpendiculars, 
will be equally distant from the three angles : or, it will be the 
centre of the circumscribing circle. 

23. Three perpendiculars drawn from the three angles of a 
triangle, upon the opposite sides, all meet in one point. 

24. Three lines bisecting the three angles of a triangle, all 
meet in one point. 

25. If d be any point in the base of a 
scalene triangle, abc: then is a b 2 . d c -f- 

AC 2 .BD = AD 2 .BC-f BCBD.DC. 




PLANE GEOMETRY : QUADRANGLES AND POLYGONS. 109 



Section III. — Of Quadrilaterals and Polygons. 



Definitions. 

1. A quadrangle or quadrilateral, is a plane figure bounded 
by four right lines. 

2. A parallelogram is a quadrangle whose opposite sides 
are parallel, as a g b h. The line a b drawn to the opposite 
corners is called the diameter or diagonal. And if two lines 
be drawn parallel to the two sides, ^ g 
through any point of the diagonal ; they 
divide it into several others, and then 
c, d are called parallelograms about the 
diameter : and e, f the complements: and 
the figure e d f a gnomon. 

3. A rectangle is a parallelogram whose sides are perpen- 
dicular to one another. 

4. A square is a rectangle of four equal sides and four equal 
angles. 

5. A rhombus is a parallelogram, 
whose sides are equal, and angles ob- 
lique. 




< 



6. A rhomboid is a parallelogram, whose sides are unequal 
and angles oblique. 



7. A trapezoid is a quadrangle, having only 
two sides parallel. 



8. A trapezium is a quadrangle, that has no 
two sides parallel. 



9. A polygon is a plane figure enclosed 
by many right lines. If all the sides and 
angles are equal, it is called a regular poly- 
gon, and denominated according to the num- 
ber of sides or angles, as a pentagon 5, a 
hexagon 6, a heptagon 7, &c. 

16 L 




I 10 PLANE GEOMETRY: QUADRILATERALS AND POLYGONS. 



10. The diagonal of a quadrangle or polygon is a line 
drawn between any two opposite corners of the figure, as 

A B. 

11. The height of a figure is a line drawn from the top per- 
pendicular to the base, or opposite side on which it stands. 

12. Like or similar figures, are those whose several angles 
are equal to one another, and the sides about the equal angles 
proportional. 

13. Homologous sides of two like figures are those between 
two angles, respectively equal. 

14. The perimeter or circumference of a figure, is the com- 
pass of it, or sum of all the lines that enclose it. 

15. The internal angles of a figure are those d 
on the inside, made by the lines that bound 
the figure, a d c b. 





16. The external angle of a figure is the 
angle made by one side of a figure, and the 
adjoining side drawn out, as b a f. 



Prop. 1. In any parallelogram the opposite sides and angles 
are equal ; and the diagonal divides it into two equal 
triangles. 

2. The diagonals of a parallelogram intersect each other in 
the middle point of both. 

3. Any line b c passing through the mid- ^J G 
die of the diagonal of a parallelogram p, 
divides the area into two equal parts. 

4. Any right line b c drawn through the middle point p of 
the diagonal of a parallelogram, is bisected in that point ; 
b p = p c. (See preceding fig.) 

5. In any parallelogram a b d c, the com- 
plements c i, and i b are equal. 

c" e D 

6. Parallelograms of equal bases and heights are equal. 

7. A parallelogram is double a triangle of the same, or an 
equal base and height. 

8. Parallelograms of the same height are to one another as 
their bases. 





PLANE GEOMETRY .* QUADRILATERALS AND POLYGONS. 1 1 1 



9. Parallelograms of equal bases are as their heights. 

10. Parallelograms are to one another, as their bases and 
heights. 

11. In any parallelogram the sum of the squares of the 
diagonals is equal to the sum of the squares of all the four 
sides. 

12. The sum of the four internal angles of any quadrilateral 
figure, is equal to four right angles. 

13. If two angles of a quadrangle be right angles, the sum 
of the other two amounts to two right angles. 

14. The sum of all tl e internal angles of a polygon is equal 
to twice as many right angles, abating four, as the polygon has 
sides. 

15. Hence all right-lined figures of the same number of sides, 
have the sum of all the internal angles equal. 

16. The sum of the external angles of any polygon is equal 
to four right angles. 

17. All right-lined figures have the sum of their external 
angles equal. 

18. In two similar figures a c, p r ; if two lines b e, q t, be 
drawn after a like manner, as suppose, ^d 
to make the angle cbe = r q t ; then 
these lines have the same proportion, as 
any two homologous sides of the figure; 
viz. be:qt::bc:qr::ab:pq::ad:ps. 

19. All similar figures are to one another as the squares of 
their homologous sides. 



20. Any figure described on the hy- 
pothenuse of a right-angled triangle, is 
equal to two similar figures described 
the same way upon the two sides :bfc 
= alc + agb. 



21. Any regular figure a b c d e, is equal 
to a triangle whose base is the perimeter 
abc d e A ; and height, the perpendicular 
o p, drawn from the centre, perpendicular to 
one side. 



22. Only three sorts of regular figures can fill up a plane 
surface, that is, the whole space round an assumed point ; and 
these are six triangles, four squares, and three hexagons. 





112 



PLANE GEOMETRY : CIRCLES, &C. 



Section IV. — Of the Circle, and Inscribed and Circum 
scribed Figures* 

Definitions. 




1. A circle is a plane figure described by 
a right line moving about a fixed point, as A , 
A c about c : or it is a figure bounded by one I 
line equidistant from a fixed point. 

2. The centre of a circle is the fixed point about which the 
line moves, c. 

3. The radius is the line that describes the circle, c A. 
Cor. All the radii of a circle are equal. 

4. The circumference is the line described by the extreme 
end of the moving line, A b d a. 



5. The diameter is a line drawn through o 
the centre, from one side to the other, a 
a D. 



6. A semicircle is half the circle, cut off by the diameter, as 

AB D. 

7. A quadrant, or quarter of a circle, is the part between two 
radii perpendicular to one another, as c d e. 

a 

B 



8. An arch is any part of the circum- 
ference A B. 



9. A sector is a part bounded by two radii, and the arch be- 
tween them, a c b. 

10. A segment is a part cut off by a right line, or cord, 

D E F, Or D A B F. 

11. A cord, a right line drawn through the circle, as d f. 






PLANE GEOMETRY : CIRCLES, &C. 113 

12. Jingle at the centre is that whose angular point is at the 
centre a c b. (See the last figure.) 



13. Jingle at the circumference is when 
the angular point is in the circumference, 
as B A D. 



14. Jingle in a segment, is the angle made by two lines 
drawn from some point of the arch of that segment to the ends 
of the base ; as b c d is an angle in the segment b c d. 

15. Jingle upon a segment, is the angle made in the oppo- 
site segment, whose sides stand upon the base of the first ; as 
bad, which stands upon the segment bcd. 

16. Ji tangent is a line touching a circle, which produced, 
does not cut it, as g a f. (Fig. to def. 5.) 

17. Circles are said to touch one another, which meet, but do 
not cut one another. 

18. Similar arches, or similar sectors, are those bounded 
by radii that make the same angle. 

19. Similar segments are those which contain similar tri- 
angles, alike placed. 

20. A figure is said to be inscribed in a circle, or a circle 
circumscribed about a figure, when all the angular points of 
the figure are in the circumference of the circle. 

21. A circle is said to be inscribed in a figure, or a figure 
circumscribed about a circle, when the circle touches all the 
sides of the figure. 

22. One figure is inscribed in another, when all the angles 
of the inscribed figure are in the sides of the other. 



Prop. 1. The radius c r, bisects any cord at 
right angles, which passes not through the 
centre, as a b. 



Cor. 1. If a line bisects a cord at right angles, it passes 
through the centre of the circle. 

Cor. 2. The radius that bisects the cord also bisects the 
arch. 

2. In a circle equal cords are equally distant from tbe 
centre. 

3. If several lines be drawn through a circle, the greatest is 
the diameter, and those that are nearer the centre are greater 
than those that are farther off. 




L 2 



114 



PLANE GEOMETRY : CIRCLES, &C. 




4. If from any point three equal right lines can be drawn to 
the circumference ; that point is the centre. 

5. No circle can cut another in more than two points 

6. There can only two equal lines be drawn from any exte- 
rior point p, to the circumference of a circle. 

7. In any circle, if several radii be drawn mak- 
ing equal angles, the arches and sectors compre- 
hended thereby will be equal, ifACB=BCD: then, 
arch a B=arch b d ; and sector a c b=b c d. 

8. In the same or equal circles, the arches, and also the sec- 
tors, are proportional to the angles intercepted by the radii. 

9. The circumferences of circles are to one another as their 
diameters. 

10. A right line, perpendicular to the diameter of a circle, 
at the extreme point, touches the circle in that point, and lies 
wholly without the circle. 

11. If two circles touch one another either inwardly or out- 
wardly, the line passing through their centres shall also pass 
through the point of contact. 



12. In a circle the angle 
at the centre is double the 
angle at the circumference, 
standing upon the same arch ; 
b d c = 2 b AC 



13. All angles in the same segment of a 
circle are equal, dac=dbc, and d g c=d h c. 



14. If the extremities of two equal arches d a, b c, De joined 

by right lines, dc,ab; they will be parallel. 



15. The angle a b c in a semicircle is a 
right angle. 





16. The angle a b g, in a greater segment a b f g, is less than 
a right angle ; and the angle a b f, in a less segment abp 
greater than a right angle. 



is 



PLANE GEOMETRY I CIRCLES, &C. 



H5 




17. If two lines cut- 
ting a circle, intersect G 
one another in a ; and H 
there be made at the cen- 

treZ_ECF=BADJ 



Then arch bd + gh=2ef, if a is within the circle ; or 
arch bd — g h = 2 e f, if a is without. 

18. If from a point without, two lines touch a circle : the 
angle made by them is equal to the angle at the centre, standing 
on half the difference, of these two parts of the circumference. 

19. The angle a = A eh d 
■j-HDG, when a is within ; or 
a=bhd — hdg, when a is 
without the circle. 

20. In a circle, the angle made at the 
point of contact between the tangent and 
any chord, is equal to the angle in the al- 




ternate segment ; 

= E G C 



e c p = e b c, and e c a 





21. A tangent to the middle point of an arch, is Darallel to 
the chord of it. 

22. If from any point b in a semicircle, 
a perpendicular b d be let fall upon the 
diameter, it will be a mean proportional 
between the segments of the diameter ; 
a d : d b : : d b : d c. A " 

23. The chord is a mean proportional between the adjoining 
segment and the diameter, from the similarity of the triangles : 
that is, ad:ab::ab:ac; and cd:cb::cb:ca. 

24. In a circle if the diameter a d be drawn, 
and from the ends of the cords a b, a c 5 per- 
pendiculars be drawn upon the diameter ; 
the squares of the chords will be as the seg- 
ments of the diameter ; a e : a p :: a b 3 : 

AC 2 . 

25. If two circles touch one another in p, 
and the line p d e be drawn through their 
centres ; and any line p a b is drawn through 
that point to cut the circles, that line will 
be divided in proportion to the diameters ; 
pa:pb::pd:pe. 




116 



PLANE GEOMETRY I CIRCLES, &C. 



26. If through any point f in the diameter 
of a circle, any chord c f d be drawn, the 
rectangle of the segments of the chord is 
equal to the rectangle of the segments of the 
diameter ;cf.fd = af.fb = also g f . 

FE. 

27. If through any point f out of the circle 
in the diameter b a produced, any line f c d 
be drawn through the circle : the rectangle of 
the whole line and the external part is equal 
to the rectangle of the whole line passing 
through the centre, and the external part ; 

DF.FC = AF.FB. 





28. Let h f be a tangent at h ; then the rectangle c f . f d =• 
square of the tangent f h. 

29. If from the same point f, two tangents be drawn to the 
circle, they will be equal ; ph = pi. 

30. If a line p f c be drawn perpendicular to the diameter 
ADof a circle, and any 
line drawn from a to cut 
the circle and the per- 
pendicular ; then the rec- 
tangle of the distances of 
the sections from a, will 
be equal to the rectan- 
gle of the diameter and 
the distance of the perpendicular from a;abxac=apxad. 

Also, a b x a c = A K a . 

31. In a circle e d f whose centre is c, and radius c E, if 
the points b a, be so placed in the 
diameter produced, that c b, c e, 
c a, be in continual proportion, then 
two lines b d, a d drawn from these 
points to any point in the circumfer- 
ence of the circle will always be in 
the given ratio of b e, to a e. 

32. In a circle, if a perpendicular d b be 
let fall from any point d, upon the diame- 
ter c I, and the tangent d o drawn from d 
then a b, a c, a o, will be continually propor- 
tional. 




PLANE GEOMETRY 



117 




33. If a triangle bdf be inscribed in a 
circle, and a perpendicular d p let fall from d 
on the opposite side b f, and the diameter 
d a drawn ; then, as the perpendicular is to 
one side including the angle d, so is the 
other side to the diameter of the circle ; 
d p : d b : : d f : d a. 

34. The rectangle of the sides of an inscribed triangle 
to the rectangle of the diameter, and the perpendicular 
third side ; b d . d f=a d.dp. 

35. If a triangle b a c be inscribed in a 
circle, and the angle a bisected by the 
right line a e d, then as one side to the 
segment of the bisecting line, within the 
triangle, so the whole bisecting line to 



is equal 
on the 



the other side 
A b . ac=b e 



a b : a e :: 

E C + A E 3 . 



AD 



a c ; and 




36. If a quadrilateral a b c d be in- 
scribed in a circle, the sum of two oppo- 
site angles is equal to two right angles ; 
a d c + a b c=two right angles. 



37. If a quadrangle be inscribed in a circle, the rectangle of 
the diagonals is equal to the sum of the rectangles of the oppo- 
site sides. 

38. A circle is equal to a triangle whose base is the circum- 
ference of the circle ; and height, its radius. 

39. The area of a circle is equal to the rectangle of half the 
circumference and half the diameter. 

40. Circles (that is, their areas) are to one another as the 
squares of their diameters, or as the squares of the radii, or as 
the squares of the circumferences. 

41. Similar polygons inscribed in circles, are to one another 
as the circles wherein they are inscribed. 

42. A circle is to any circumscribed rectilineal figure, as the 
circle's periphery to the periphery of the figure. 

43. If an equilateral triangle a b c be in- 
scribed in a circle ; the square of the side 
thereof is equal to three times the square of 
the radius : a b 3 = 3 a d 8 . 



17 




118 



PLANE GEOMETRY I CIRCLES, &C. 



44. A square inscribed in a circle, is equal to twice the 
square of the radius. 



45. The side of a regular hexagon inscribed 
in a circle, is equal to the radius of the circle ; 
B e= b c. 



46. If two chords in a circle mutually inter- 
sect at right angles, the sum of the squares of 
the segments of the chords is equal to the 
square of the diameter of the circle, a p a + 
p b 2 4- p c 3 + p i> 2 = diam. 3 




47. If the diameter pq be divided into two 
parts at any point r, and if r s be drawn perpen- 
dicular to pq; also r t applied equal to the 
radius, and t r produced to the circumference 
at v : then, between the two segments p r, 

R Q, 

r t is the arithmetical mean, 
r s is the geometrical mean, 
r v is the harmonical mean. 




48. If the arcs p q, q r, r s, &c. be 
equal, and there be drawn the chords 
p q, p r, p s, &c. then it will be p q : p r : : 
pr :pq+ ps::ps :pr +pt::pt : 

PS + PV, &c. 




PLANE GEOMETRY: CIRCLES, &C, 



119 



49. The centre of a circle being o, and 
p a point in the radius, or in the radius 
produced ; if the circumference be divided 
into as many equal parts a b, b c, c d, &c. 
as there are units in 2 n, and lines be 
drawn from p to all the points of division ; 
then shall the continual product of all the 
alternate lines, viz. paxpcXpe &c. 
be = r n — x n when p is within the 
circle, or = x n — r n when p is 
without the circle ; and the product 
of the rest of the lines, viz. p b x p d 
X pf, &c. = r n + x n : where r = 

a o the radius, and x = o p the dis- 
tance of p from the centre. 

50. A circle may thus be divided into any 
number of parts that shall be equal to one 
another both in area and perimeter. Divide 
the diameter q r into the same number of 
equal parts at the points s, t, v, &c. ; then 
on one side of the diameter describe semi- 
circles on the diameters q s, q t, q v, and on 
the other side of it describe semicircles on 
r v, r t, r s ; so shall the parts, 1 7, 3 5, 
5 3, 7 1, be all equal, both in area and peri- 
meter. 





Section V. — Of Planes and Solids. 



Definitions. 



1. The common section of two planes, is the line in which 
they meet, or cut each other. 

2. A line is perpendicular to a plane, when it is perpendi- 
cular to every line in that plane which meets it. 

3. One plane is perpendicular to another, when every line 
of the one, which is perpendicular to the line of their common 
section, is perpendicular to the other. 



120 



SOLID GEOMETRY. 



4. The inclination of one plane to another, or the angle they 
form between them, is the angle contained by two lines, drawn 
from any point in the common section, and at right angles to the 
same, one of these lines in each plane. 

5. Parallel planes are such as being produced ever so far 
both ways, will never meet, or which are everywhere at an 
equal perpendicular distance. 

6. A solid angle is that which is made by three or more plane 
angles, meeting each other in the same point. 

7. Similar solids, contained by plane figures, are such as 
have all their solid angles equal, each to each, and are bounded 
by the same number of similar planes, alike placed. 

S. A prism is a solid whose ends are parallel, equal, and like 
plane figures, and its sides, connecting those ends, are parallelo- 
grams. 

9. A prism takes particular names according to the figure of 
its base or ends, whether triangular, square, rectangular, penta- 
gonal, hexagonal, &c. 

10. A right or upright prism, is that which has the planes 
of the sides perpendicular to the planes of the ends or base. 

11. A parallelopiped, or a parallelopipedon, 
is a prism bounded by six parallelograms, every 
opposite two of which are equal, alike, and 
parallel. 

12. A rectangular parallelopipedon is that whose bound- 
ing planes are all rectangles, which are perpendicular to each 
other 



\ \ 


1 


N \ 



13. A cube is a square prism, being bounded /L. 
by six equal square sides or faces, which are per- 
pendicular to each other. 



71 



14. A cylinder is a round prism having circles for 
its ends ; and is conceived to be formed by the rota- 
tion of a right line about the circumferences of two 
equal and parallel circles, always parallel to the axis. 



15. The axis of a cylinder is the right line joining the cen- 
tres of the two parallel circles about which the figure is de- 
scribed. 





SOLID GEOMETRY. 121 

16. A Pyramid is a solid whose base is any right- 
lined plane figure, and its sides triangles, having all 
their vertices meeting together in a point above the 
base, called the vertex of the pyramid. 

17. Pyramids, like prisms, take particular names from the 
figure of their base. 

18. A cone is a round pyramid having a circular 
base, and is conceived to be generated by the rota- 
tion of a right line about the circumference of a circle, 
one end of which is fixed at a point above the plane of 
that circle. 

19. The axis of a cone is the right line, joining the vertex 
or fixed point, and the centre of the circle about which the 
figure is described. 

20. Similar cones and cylinders, are such as have their al- 
titudes and the diameters of their bases proportional. 

21. A sphere is a solid bounded by one curve surface, which 
is every where equally distant from a certain point within, 
called the centre. It is conceived to be generated by the rota- 
tion of a semicircle about its diameter, which remains fixed. 

22. The axis of a sphere is the right line about which the 
semicircle revolves, and the centre is the same as that of the 
revolving semicircle. 

23. The diameter of a sphere is any right line passing 
through the centre, and terminated both ways by the surface. 

24. The altitude of a solid is the perpendicular drawn from 
the vertex to the opposite side or base. 

Prop. 1. If any prism be cut by a plane parallel to its base, 
the section will be equal and like to the base. 

2. If a cylinder be cut by a plane parallel to its base, the 
section will be a circle, equal to the base. 

3. All prisms and cylinders, of equal bases and altitudes, 
are equal to each other. 

4. Rectangular parallelopipedons, of equal altitudes, are to 
each other as their bases. 

5. Rectangular parallelopipedons, of equal bases, are to each 
other as their altitudes. 

6. Because, prisms and cylinders are as their altitudes, when 
their bases are equal : and, as their bases when their altitudes 
are equal. Therefore, universally, when neither are equal, 
they are to one another as the product of their bases and alti- 
tudes : hence, also, these products are the proper numeral 
measures of their quantities or magnitudes. 



M 



122 SOLID GEOMETRY. 

7. Similar prisms and cylinders are to each other as the cubes 
of their altitudes, or of any like linear dimensions. 

8. In any pyramid a section parallel to the base is similar to 
the base ; and these two planes are to each other as the squares 
of their distances from the vertex. 

9. In a cone, any section parallel to the base is a circle ; and 
this section is to the base as the squares of their distances from 
the vertex. 

10. All pyramids and cones, of equal bases and altitudes, 
are equal to one another. 

11. Every pyramid is a third part of a prism of the same 
base and altitude. 

12. If a sphere be cut by a plane, the section will be a 
circle. 

13. Every sphere is two-thirds of its circumscribing cy- 
linder. 

14. A cone hemisphere, and cylinder of the same base and 
altitude, are to each other as the numbers 1, 2, 3. 

15. All spheres are to each other as the cubes of their diame- 
ters; all these being like parts of their circumscribing cylinders. 

16. None but three sorts of regular plane figures joined 
together can make a solid angle : and these are, 3, 4, or 5 
triangles, 3 squares, and three pentagons. 

And therefore there can only be five regular bodies, the py- 
ramid, cube, octaedron, dodecaedron, and icosaedron. 

17. No other but only one sort of the five regular bodies, 
joined at their angles, can completely fill a solid space ; viz. 
eight cubes. 



18. A sphere is to any circumscribing 
solid b f, (all whose planes touch the 
sphere) ; as the surface of the sphere to the 
surface of the solid. 




E 

19. All bodies circumscribing the same sphere, are to one 
another as their surfaces. 

20. The sphere is the greatest or most capacious of all 
bodies of equal surface. 



PRACTICAL GEOMETRY. 



123 



Section VI. — Practical Geometry. 

It is not intended in this place to present a complete collec- 
tion of Geometrical Problems, but merely a selection of the most 
useful, epecially in reference to the employments of mechanics 
and engineers. 

The instruments well known to be used in geometrical con- 
structions, are the scale and compasses, the semicircular or the 
circular protractor, the sector, and a parallel ruler. To these 
a few other useful instruments may be added, which we shall 
describe as we proceed ; speaking first of the Triangle and 
Ruler. 

These are, as their names indicate, a triangle, that is to say, 
an isosceles right-angled triangle, and a ruler, both made of 
well seasoned wood, or of ivory, ebony, or metal. Each side 
a b, a c, of the triangle, about the right angle a, being 3, 4, 6, 
or 8 inches, according to the magnitude of the figures, in 
whose construction it is a 

likely to be employed. 
About the middle of the 
triangle there should be a 
circular orifice, as shown 
in the figures ; and if a 
scale of equal parts be 
placed along each of the three sides, all the better. The ruler 
may be from 12 to 18 inches in length ; and it also may, use- 
fully, have a scale along one of its sides. The conjoined appli- 
cation of these instruments is of great utility ; as will soon 
appear. 

Prob. I. To bisect a given line. 

Let a b be the line proposed. Lay the longest side b c of 
the triangle so as to coincide with a b, and so that its angle b 
shall coincide with the point a ; and along the side b a of the 
triangle draw a line a d. Then slide the 
base b c of the triangle along the line 
a by until c coincides with b, and draw 
in coincidence with the side c A, the line 
b d intersecting the former in d. Next 
bring the ruler to coincide with a b, and 
in contact with it lay one of the legs b a of the triangle ; then 
slide the triangle along the ruler, until the other leg A c passes 
through the point d : draw along a c, so posited, the line d i ; 
it will be perpendicular to a b, and will bisect it in i, the point 
required. 





124 



PRACTICAL GEOMETRY. 



Prob. 2. Through a given point, c, to draw a line parallel to 
a given line a b. 




Place one of the sides of the triangle in contact with the line 
a B. Lay the ruler against one of the other sides of the tri- 
angle ; and keeping it steady, slide along the triangle until 
the same side which had been made to coincide with part of 
the line a b touches the point c : then, along that side, draw 
through c the line e f ; it will be parallel to a b as required. 

Prob. 3. To bisect a given angle : then to bisect its half ; 
and so on 




Let bap be the proposed angle. Through any point b 
draw b e parallel to a p (by the former problem). Upon b e 
setoff, with the compasses, from the scale at the edge of the 
ruler, b c = b a : join a c ; it will bisect the angle bap. 

Again, set off, upon b e, from c, cd=ca: join a d ; it 
will bisect cap, or quadrisect bap. 

Again, set off, upon b e, d e = d a : join e a ; so shall 
eap be-g- of bap: and so on. 

Prob. 4. To erect a perpendicular at any given point c, in 
a given line a b. 

1st Method. Apply one of the legs, e f, of the triangle, 
upon the line a b. Lay the side of the ruler h i, against the hypo 
thenuse, e g, of the triangle, and, keeping it steady, slide 
the triangle upwards until the side f g touches the point c. 



PRACTICAL GEOMETRY. 



126 



Then draw c d in contact with that leg, and it will be the per 
pendicular required. 




2d Method. Apply the hypothenuse, g e, of the triangle to 
the line a b. Lay the edge of the ruler h i, against the leg 
g e. Keep it steady, and turn the triangle so that the other 
leg f e may be laid against the ruler. Then slide the trian- 
gle upwards until the hypothenuse touches the point c : then 
in coincidence with it draw c d, and it will be the perpendicu- 
lar required. 

Note. — After similar methods may a perpendicular be let fall 
from a point d above a line a b upon it. 

3d Method, by a ruler and compasses only : as suppose it were 
required to cut the end of a plank square. Let abcd be the 
plank, of which the end b d is required to be squared. The 
edge A b being quite straight, 
open the compasses to any con- 
venient distance, and place the 
point of one leg at b, and the 
other at any point as f. Keep one 
leg at f, and turn the other round 
till it touches the edge a b at e ; 
keep them firm, and apply the 
straight edge to e f, as the figure 
shows ; keep the leg still at f, and turn them over into the posi- 
tion f G, g being close to the straight edge, and make a mark at g. 
Now, if the straight edge be applied to g and b, and g b be 
drawn, it will be square to the edge a b. 

Note. — In this construction it is evident that f is the centre, 
and e g the diameter of a semicircle that passes through b ; con- 
quently b is a right angle. 

18 m2 





126 PRACTICAL GEOMETRY. 

Prob. 5. To divide a given line a b into any proposed num- 
ber of equal parts. 

1st Method. Draw any other line a c, forming any angle with 
the given line ab : on which set off as many of 
any equal parts, ad,de,ef,fc, as the line A b 
is to be divided into. Join b c ; parallel to which 
draw the other lines fg, eh,di: then these a i h g 
will divide a b in the manner as required. 

2d Method, without drawing parallel lines. Let A b be the 
line which is to be divided into n equal parts. Through one 
extremity a draw any right line a c, 
upon which set off n + 1 equal parts, 
the point d being at the termination of 
the (n-fl)th part. Join d b and pro- 
duce it until the prolongation be = b d. 
Let f be the termination of the 
(n— l)th part. Join p e, and the right 
line of junction will cut the given line 
a b in the point p, such that b p=„ a b; and of course distances 
equal to bp set off upon b a, will divide it, as required * 

Prob. 6. At a given point a, in a given line a b, to make an 
angle equal to a given anglte c. 

From the centres a and c, with any one ra- 
dius, describe the arcs d e, f g. Then, with 
radius d e, and centre f, describe an arc cutting 
f G in g. Through g draw the line A g ; and 
it will form the angle required. 





F b 



Prob. 7. To find the centre of a circle. 



Draw any chord a b, and bisect it perpendicu- 
larly with the line c d. Then bisect c d in o, the 
centre required. 



* The truth of this method is easily demonstrated. Through i the intermediate 
point of division, on a c, between f and d, draw i b. Then, because d b = b e 
and d i = i r, ib is parallel to f p. Consequently, bp:ba::if:ia::1:«, 
by construction. 




PRACTICAL GEOMETRY. 



127 



Prob. 8. To describe the circumference of a circle through 
three given points, a, b, c. 

From the middle point b draw chords b a, 
b c, to the other two points, and bisect these 
chords perpendicularly by lines meeting in o, 
which will be the centre. Then from the 
centre o, at the distance of any of the points, 
as o a, describe a circle, and it will pass 
through the two other points B,c,as required. 





Prob. 9. On a given chord a b to describe an arc of a circle 
that shall contain any number of degrees ; performing the ope- 
ration without compasses, and without finding the centre of the 
circle. 

Place two rulers, forming an angle a c b, equal to the supple- 
ment of half the given number of degrees, and fix them in c. 
Place two pins at the extremities of the given chord, and hold 
a pencil in c ; 
then move the 
edges of this in- 
strument against 
the pins, and the 
pencil will describe the arc required. 

Suppose it is required to describe an arc of 50 degrees on the 
given chord a b ; subtract 25 degrees (which is half the given 
angle) from 180, and the difference, 155 degrees, will be the sup- 
plement. Then form an angle a c b of 155° with the two rulers, 
and proceed as has been shown above. 

Prob. 10. To describe mechanically the circumference of a 
circle, through three given points, a, b, c, when the centre is 
inaccessible ; or the circle too large to be described with com- 
passes. 

Place two rulers, m n, r s, crossways, touching the three 
points abc. Fix them in v by a pin, and by a transverse piece 
t. Hold a pencil in a, and describe the 
arc b a c, by moving the angle r a n, so 
as to keep the outside edges of the rulers 
against the pins b c. Remove the instru- 
ment r v n, and on the arc described 
mark two points, d, e, so that their dis- 
tance shall be equal to the length b c. 
Apply the edges of the instrument 
against d e, and with a pencil in g de- 
scribe the arc b c, which will complete 
the circumference of the circle required. 




128 



PRACTICAL GEOMETRY. 




Otherwise. — Let an axle of 12 or 15 inches long carry two 
unequal wheels a and b, of which one, a, shall be fixed, while 
the other, b, shall be susceptible of motion along the axle, and 
then placed at any assigned distance, a, b upon the paper or 
plane, on which the circle is to be described. Then will a and 
b be analogous to the ends of a conic fru strum, the vertex of 
the complete cone being the centre of the circle which will be 
described by the rim, or edge, 
of the wheel a, as it rolls upon 
the proposed plane. Then, it 
will be, as the diameter of 
the wheel a, is to the differ- 
ence of the diameters of a and b, so is the radius of the circle 
proposed to be described by a, to the distance A b, at which 
the two wheels must be asunder, measured upon the plane on 
which the circle is to be described. 

The wheel b will evidently describe, simultaneously, another 
circle, whose radius will be less than that of the former by a b. 

Prob. 11. To divide any given angle a b c into three equal 
parts. 

From b, with any radius, describe 
the circle acda. Bisect the angle abc 
by b e, and produce a b to d. On the 
edge of a ruler mark off the length 
of the radius a b. Lay the ruler on 
d, and move it till one of the marks 
on the edge intersects b e, and the 
other the arc Ac in g. Set off the 
distance c G from g to f : and draw 
the lines b f, and b g, they will tri- 
sect the angle abc. 

Otherwise, by means of Mr. R. Christie's ingenious instru- 
ment for the mechanical trisection of an angle. 




PRACTICAL GEOMETRY. 129 



This instrument may be made either of wood or metal. Fig. 
1 represents it applied to, and trisecting the angle hcb, and 
fig. 2 represents it shut up. The pieces h c, e c, f c, and 
g c, are all of the same length, and moveable on the joint c. 
The joints a, b, c, and d, are all equally distant from c. The 
connecting pieces a e, e b, b h, h c, c i, and i d, are all equal ; 
and the pieces ef,fh,hg, and g i, are equal to each other, but 
longer than the preceding pieces. Two sockets,/ and g, fit, 
and move up or down on the pieces e c and f c. The pieces are 
all connected by pivots at the joints, represented by the small 
letters a, b, c, d, &c, and the connecting pieces fit in between 
the other when the apparatus is shut. 

In applying this instrument it is only necessary to lay the 
centre c on the vertex of the given angle, c g, on one of the 
sides forming it, and to move h c till it coincides with the other; 
then each of the angles, hce,ecp, and p c g, will be a third 
of the angle h c g. For it is manifest, that the angle hce can- 
not be increased without increasing the angle a e b, and that 
a e b cannot be increased without diminishing the angle b e f 
and the distance fb. But because b e is equal to b h,f e to/ A, 
and f b common to the two triangles f e b and f h b ; the an- 
gle f h b must be always equal to the angle fe b, and conse- 
quently b h c to a e b; therefore hce must in all positions of 
the apparatus, continue equal to e c f. In the same manner it 
might be shown that the angles e c f and f c g will always con- 
tinue equal. Hence the angle hcb has been trisected by the 
straight lines e c and f c. If the instrument had been applied 
to the angle d c b, it would have taken the position represented 
by the dotted lines. 

Note. — It is evident that instruments may be made on the 
same principle to divide an angle into any other number of equal 
parts. 



130 



PRACTICAL GEOMETRY. 



Prob. 12. To cut off from a given line a b, supposed to be 
very short, any proportional part. 

Suppose, for example, it were required to find the -j- 1 ^, T 2 ^., J J9 
&c. of the line a b in the first figure below. From the ends a 
and b draw adbc perpendicular to a b. From a to d set off 
any opening of the compasses 12 times, and the same from b 
to c. Through the divisions 1,2, 3, &c. draw lines 1 /, 2 g, 
&c. parallel to a b. Draw the diagonal a c, and 1 d will be 
the T L of a,b; 2 c, T \, and so on. The same method is applica- 
ble to any other part of a given line. 




"run 


1 


j 




-A- 


j 


1 1 ■ 


_4Z±L 




ii.,. 


JJ.Jj 




riTiliQ . 




Ml 1 1 






I 




















1 








M 


,LB- I 



D C 



Prob 13. To make a diagonal scale, say of feet, inches, and 
tenths of an inch. 

Draw an indefinite line a b, on which set off from a to b the 
given length for one foot, any required number of times. From 
the divisions a, c, h, b, draw a d, c e, &c. perpendicular to a b. 
On a d and b f set off any length ten times ; through these di- 
visions draw lines parallel to a b. Divide a c and d e into 12 
equal parts, each of which will be one inch. Draw the lines a 1, 
g 2, &c. and they will form the scale required ; viz. each of the 
larger divisions from e to 1, 1 to 2, &c. will represent a foot ; 
each of the twelve divisions between d and e an inch; and the 
several perpendiculars parallel to r c in the triangle e c r, t \j-, 
to 9 tu> &c. of an inch. 

Note. — If the scale be meant to represent feet, or any other 
unit, and tenths and hundredths, then d e must be divided into 
ten instead of twelve equal parts. 

Prob. 14. Given the side of a regular polygon of any number 
of sides, to find the radius of the circle in which it may be in- 
scribed. 

Multiply the given side of the polygon by the number which 
stands opposite the given number of sides in the column enti- 
tled radius of circum. circle; the product will be the radius 
required. 



PRACTICAL GEOMETRY. 



131 



Thus, suppose the polygon was to be an octagon, and each 
side 12, then 1-3065628 X 12 = 15-6687536 would be the radius 
sought. Take 15-67 as a radius from a diagonal scale, describe 
a circle, and from the same scale, taking off 12, it may be applied 
as the side of an octagon in that circle. 

Prob. 15. Given the radius of a circle to find the side of any 
regular polygon (sides not exceeding 12) inscribed in it. 

Multiply the given radius by the number in the column en- 
titled factors for sides, standing opposite the number of the 
proposed polygon; the product is the side required. 

Thus, suppose the radius of the circle to be 5, then 
5 x 1-732051 = 8-66025, will be the side of the inscribed equila- 
teral triangle. 

TABLE OF POLYGONS. 



O W 

3 


Names. 


Multipliers 
for areas. 


Radius of 
circum.cir. 


Factors 
for sides. 


Trigon 


0-4330127 


0-5773503 


1-732051 


4 


Tetragon, or Square 


1-0000000 


0-7071068 


1-414214 


5 


Pentagon 


1-7204774 


0-8506508 


1-175570 


6 


Hexagon 


2-5980762 


1-0000000 


1-000000 


7 


Heptagon 


3-6339124 


1-1523824 


0.867767 


8 


Octagon 


4-8284271 


1-3065628 


0-765367 


9 


Nonagon 


6-1818242 


1-4619022 


0-684040 


10 


Decagon 


7-6942088 


1-6180340 


0-618034 


11 


Undecagon 


9-3656399 


1-7747324 


0-563465 


12 


Dodecagon 


11-1961524 


1-9318517 


0-517638 



Prob. 16. To reduce a rectilinear figure of 6, 7, or more sides, 
to a triangle of equal area. 

This is a very useful problem, as it saves much labour in com- 
putation. 

Suppose abcdef g to be the proposed space to be reduced 
to a triangle. Lay a parallel ruler from a to c, and move 
it until it pass through b, marking the point 1 in which it 
cuts a G continued. Then lay the ruler 
through 1 and d, and move it until it pass 
through c, and mark the point 2 where it 
cuts a G. Next lay the ruler through 2 
and f, move it up till it pass through d, 
marking the point 3 where it cuts a g 
continued. Again, lay the ruler through 
3 and f, move it up until it pass through e, and mark 4, 
the point of intersection with a a produced. Lastly, draw 





132 PRACTICAL GEOMETRY. 

the right line 4 f ; so shall the triangle 4 f g be equal in area 
to the irregular polygon a b c d e f g. 

Here b 1 is parallel to a c ; so that if c 1 were drawn, the 
triangle a 1 c would be equal to a c b : and by the mechanical 
process this reduction is effected. In like manner, the other 
triangles are referred, one by one, to equal triangles, having 
their bases on g a or its prolongation. Hence the principle of 
the reduction is obvious. 

Prob. 17. To reduce a simple rectilinear figure to a similar 
one upon either a smaller or a larger scale. 

Pitch upon a point p any where 
about the given figure abode, 
either within it, or without it, or 
in one side or angle; but near the 
middle is best. From that point p 
draw lines through all the angles; 
upon one of which take p a to p a 
in the proposed proportion of the 
scales, or linear dimensions ; then draw a b parallel to A b, b c 
to b c, &c. ; so shall ab cdebe the reduced figures sought, either 
greater or smaller than the original. (Hutton's Mens.) 

Otherwise to reduce a Figure by a Scale. — Measure all the 
sides and diagonals of the figure, as a b c d e, by a scale; and 
lay down the same measures respectively from another scale, in 
the proportion required. 

To reduce a Map, Design, or Figure, by Squares. — Divide 
the original into a number of little squares, and divide a fresh 
paper, of the dimensions required, into the same number of other 
squares, either greater or smaller, as required. This done, in 
every square of the second figure, draw what is found in the cor- 
responding square of the first or original figure. 

The cross lines forming these squares may be drawn with a 
pencil, and rubbed out again after the work is finished. But a 
more ready and convenient way, especially when such reduc- 
tions are often wanted, would be to keep always at hand frames 
of squares ready made, of several sizes; for by only just laying 
them down upon the papers, the corresponding parts may be 
readily copied. These frames may be made of four stiff or in- 
flexible bars, strung across with horse hairs, or fine catgut. 

When figures are rather complex, the reduction to a different 
scale will be best accomplished by means of such an instrument 
as Professor Wallace's Eidograph, or by means of a Panto- 
graph, an instrument which is now considerably improved by 
simply changing the place of the fulcrum. See the Mechanics' 
Oracle, part II. page 33. 



PLANE TRIGONOMETRY. 133 



CHAPTER IV. 
TRIGONOMETRY. 

Section I. — Plane Trigonometry. 

1. Plane Trigonometry is that branch of mathematics by 
which we learn how to determine or compute three of the six 
parts of a plane, or rectilinear triangle, from the other three, 
when that is possible. 

The determination of the mutual relation of the sines, tan- 
gents, secants, &c. of the sums, differences, multiples, &c. of arcs 
or angles; or the investigation of the connected formulae, is, also, 
usually classed under plane trigonometry. 

2. Let a c b be a rectilinear angle: if about c as a centre, with 
any radius, c A, a circle be described, intersecting c a, c b, in 
A, b, the arc a b is called the measure of the angle acb. (See 
the next figure.) 

3. The circumference of a circle is supposed to be divided 
or to be divisible into 360 equal parts, called degrees; each de- 
gree into 60 equal parts, called minutes; each of these into 60 
equal parts, called seconds ; and so on to the minutest possible 
subdivisions. Of these, the first is indicated by a small circle, 
the second by a single accent, the third by a double accent, &c. 
Thus, 47° 18' 34" 45'", denotes 47 degrees, 18 minutes, 34 
seconds, and 45 thirds. So many degrees, minutes, seconds, &c. 
as are contained in any arc, of so many degrees, minutes, seconds, 
&c. is the angle of which that arc is the measure said to be. 
Thus, since a quadrant, or quarter of a circle, contains 90 de- 
grees, and a quadrantal arc is the measure of a right angle, a 
right angle is said to be one of 90 degrees. 

4. The complement of an arc is its difference from a quad- 
rant; and the complement of an angle is its difference from a 
right angle. 

5. The supplement of an arc is its difference from a semicir- 
cle, and the supplement of an angle is its difference from two 
right angles. 

6. The sine of an arc is a perpendicular let fall from one ex- 
tremity upon a diameter passing through the other. 

19 N 



134 



PLANE TRIGONOMETRY. 



7. The versed sine of an arc is that part of the diametei 
which is intercepted between the foot of the sine and the arc. 

8. The tangent of an arc is a right line which touches it in 
one extremity, and is limited by a right line drawn from the 
centre of the circle through the other extremity. 

9. The secant of an arc is a sloping line which thus limits 
the tangent. 

10. These are also, by way of accommodation, said to be the 
sine, tangent, &c. of the angle measured by the aforesaid arc, 
to its determinate radius. 

11. The cosine of an arc or angle, is the sine of the comple- 
ment of that arc or angle: the cotangent of an arc or angle is 
the tangent of the complement of that arc or angle. The co- 
versed sine and co-secant are defined similarly. 

To exemplify these definitions by the annexed diagram : let 
a b be an assumed arc of a circle described with the radius a c, 
and let a e be a quadrantal arc ; let b d be demitted perpen- 
dicularly from the extremity b upon 
the diameter a a'; parallel to it let 
a t be drawn, and limited by c t: let 
g b and e m be drawn parallel to a a', 
the latter being limited by c t or c T 
produced. Then b e is the comple- 
ment of b a, and angle bce the com- 
plement of angle bca; b e a' is the 

supplement of b a, and angle b c a' the supplement of b c A ; 
b d is the sine, d a the versed rine, a t the tangent, c t the 
secant, g b the cosine, a e the coversed sine, e m the cotangent, 
and c m the cosecant, of the arc a e, or, by convention, of the 
angle acb. 

Note. — These terms are indicated by obvious contractions : 
Thus, for sine of the arc a b we use sin a b, 





E 




M 






^B , 






G 


\ 










fPr 




A'[ D ' 




^^ 










c "- 


1 




T'P^ 




/p 




B^s 











tangent . . 


ditto . 


. . tan a b, 


secant . . . 


ditto . 


. . sec A B, 


versed sine 


ditto . 


. versin a b, 


cosine . . . 


ditto . 


. . COSAB, 


cotangent . 


ditto . 


. COt A B, 


cosecant 


ditto . 


. cosec A B, 


coversed sine 


i ditto . 


. coversin a b. 



PLANE TRIGONOMETRY. 135 



Corollaries from the above Definitions. 

12. (a.) Of any arc less than a quadrant, the arc is less than 
its corresponding tangent : and of any arc whatever, the chord 
is less than the arc, and the sine less than the chord. 

(b.) The sine b d of an arc a b, is half the chord b f of the 
double arc bap. 

(c.) An arc and its supplement have the same sine, tangent, 
and secant. (The two latter, however, are affected by different 
signs, + or — 5 according as they appertain to marks less or 
greater than a quadrant.) 

(d.) When the arc is evanescent, the sine, tangent, and versed 
sine, are evanescent also, and the secant becomes equal to the 
radius, being its minimum limit. As the arc increases from 
this state, the sines, tangents, secants, and versed sines increase ; 
thus they continue till the arc becomes equal to a quadrant a e, 
and then the sine is in its maximum state, being equal to radius, 
thence called the sine total; the versed sine is also then equal 
to the radius: and the secant and tangent becoming incapable 
of mutually limiting each other, are regarded as infinite. 

(e.) The versed sine of an arc, together with its cosine, are 
equal to the radius. Thus, a d+b g=a d + d c = a c. (This is 
not restricted to arcs less than a quadrant.) 

(f.) The radius, tangent, and secant, constitute a right-angled 
triangle cat. The cosine, sine, and radius, constitute another 
right-angled triangle c d b, similar to the former. So, again, 
the cotangent, radius, and cosecant, constitute a third right- 
angled triangle, m e c, similar to both the preceding. Hence, 
when the sine and radius are known, the cosine is determined 
by the property of the right-angled triangle. 

The same may be said of the determination of the secant, from 
the tangent and radius, &c. &c. &c. 

(g.) Further, since c d : d b : : c a : a t, we see that the tan- 
gent is a fourth proportional to the cosine, sine and radius. 

Also, cd:cb::ca:ct; that is, the secant is a third pro- 
portional to the cosine and radius. 

Again, cg: gb::ce:em ; that is, the cotangent is a fourth 
proportional to the sine, cosine, and radius. 

And bd:bc::ce:cm; that is, the cosecant is a third pro- 
portional to the sine and radius. 



136 PLANE TRIGONOMETRY. 

(h.) Thus, employing the usual abbreviations, we should 
have 

1. cos= ^(rad 3 — sin 3 ). 2. tan= >/(sec 3 — rad s ). 

3. sec = -v/(rad 3 + tan 3 ). 4. cosec= >/(rad a +cot 3 ). 

5. tan = radXsin =I^!. 6. eot ^adxcos ^ 
cos cot sin tan 

7. sec=E^!. 8. cosec=!^!. 

cos sin 

These, when unity is regarded as the radius of the circle, 
become 

1. cos= <v/(l — sin 3 ). 2. tan= v'Csec 3 — 1). 

3. sec= y/(l + tan 3 ). 4. cosec =V(l+cot 3 ). 

5. tan= ™L.= _L. 6. cot= £21= JL. 7. sec= -L. 
cos cot sin tan cos 

8. cosec= _L . 

sin 

13. From these, and other properties, and theorems, mathe- 
maticians have computed the lengths of the sines, tangents, 
secants, and versed sines, to an assumed radius, that corres- 
pond to arcs from one second of a degree, through all the 
gradations of magnitude, up to a quadrant, or 90°. The results 
of the computations are arranged in tables called Trigonome- 
trical Tables for use. The arrangement is generally appro- 
priated to two distinct kinds of these artificial numbers, 
classed in their regular order upon pages that face each other. 
On the left hand pages are placed the sines, tangents, secants, 
&c. adapted at least to every degree, and minute, in the quad- 
rant, computed to the radius 1, and expressed decimally. 
On the right hand pages are placed in succession the corres- 
ponding logarithms of the numbers that denote the several 
sines, tangents, &c. on the respective opposite pages. Only, 
that the necessity of using negative indices in the logarithms 
may be precluded, they are supposed to be the logarithms of 
sines, tangents, secants, &c. computed to the radius 10000000000. 
The numbers thus computed and placed on the successive right 
hand pages are called logarithmic sines, tangents, &c. The 
numbers of which these are the logarithms, and which are 
arranged on the left hand pages, are called natural sines, tan- 
gents, &c. The tables of Hutton, Galbraith, Ursin, and 
Young, will serve well for the usual purposes : if very accurate 
computations occur, in which the sines, tangents, &c. are re- 
quired to seconds, the tables of Taylor and Bagay may be ad- 
vantageously consulted. 



PLANE TRIGONOMETRY. 137 



II. — General Properties and Mutual Relations, 

1. The chord of any arc is a mean proportional between the 
versed sine of that arc and the diameter of the circle. 

2. As radius, to the cosine of any arc; so is twice the sine of 
that arc, to the sine of double the arc. 

3. The secant of any arc is equal to the sum of its tangent, 
and the tangent of half its complement. 

4. The sum of the tangent and secant of any arc, is equal to 
the tangent of an arc exceeding that by half its complement. 
Or, the sum of the tangent and secant of an arc is equal to the 
tangent of 45° plus half the arc. 

5. The chord of 60° is equal to the radius of the circle ; the 
versed sine and cosine of 60° are each equal to half the radius, 
and the secant of 60° is equal to double the radius. 

6. The tangent of 45° is equal to the radius. 

7. The square of the sine of half any arc or angle is equal to 
a rectangle under half the radius and the versed sine of the 
whole ; and the square of its cosine, equal to a rectangle under 
half the radius and the versed sine of the supplement of the 
whole arc or angle. 

8. The rectangle under the radius and the sine of the sum or 
the difference of two arcs is equal to the sum of the difference 
of the rectangles under their alternate sines and cosines. 

9. The rectangle under the radius and the cosine of the sum 
or the difference of two arcs, is equal to the difference or the sum 
of the rectangles under their respective cosines and sines. 

10. As the difference or sum of the square of the radius and 
the rectangle under the tangents of two arcs, is to the square of 
the radius; so is the sum or difference of their tangents, to the 
tangent of the sum or difference of the arcs. 

11. As the sum of the sines of two unequal arcs, is to their 
difference ; so is the tangent of half the sum of those two arcs 
to the tangent of half their difference. 

12. Of any three equidifferent arcs, it will be as radius, to the 
cosine of their common difference, so is the sine of the mean 
arc, to half the sum of the sines of the extremes; and, as radius 
to the sine of the common difference, so is the cosine of the 
mean arc to half the difference of the sines of the two extremes. 

(a.) If the sine of the mean of three equidifferent arcs 

N 2 



138 PLANE TRIGONOMETRY. 

dius being unity) be multiplied into twice the cosine of the com- 
mon difference, and the sine of either extreme be deducted from 
the product, the remainder will be the sine of the other ex- 
treme. 

(b.) The sine of any arc above 60°, is equal to the sine of ano- 
ther arc as much below 60°, together with the sine of its excess 
above 60°. 

Remark. From this latter proposition, the sines below 60° 
being known, those of arcs above 60° are determinable by addi- 
tion only. 

13. In any right-angled triangle, the hypothenuse is to one 
of the legs, as the radius to the sine of the angle opposite to that 
leg ; and one of the legs is to the other as the radius to the tan- 
gent of the angle opposite to the latter. 

14. In any plane triangle, as one of the sides is to another, so 
is the sine of the angle opposite to the former to the sine of the 
angle opposite to the latter. 

15. In any plane triangle it will be, as the sum of the sides 
about the vertical angle is to their difference, so is the tangent 
of half the sum of the angles at the base, to the tangent of half 
their difference. 

16. In any plane triangle it will be, as the cosine of the dif- 
ference of the angles at the base, is to the cosine of half their 
sum, so is the sum of the sides about the vertical angles to the 
third side. Also, as the sine of half the difference of the angles 
at the base, is to the sine of half their sum, so is the difference 
of the sides about the vertical angle to the third side, or base.* 

17. In any plane triangle it will be, as the base, to the sum 
of the two other sides, so is the difference of those sides to the 
difference of the segments of the base made by a perpendicular 
let fall from the vertical angle. 

18. In any plane triangle it will be, as twice the rectangle 
under any two sides, is to the difference of the sum of the 
squares of those two sides and the square of the base, so is 
the radius to the cosine of the angle contained by the two 
sides. 

Cor. When unity is assumed as radius, then if a c, a b, 
b c, are the sides of a triangle, this prop, gives cos. c = 

A C 3 -f-B C 2 A B 2 

: and similar expressions for the other an- 

2 c b.c a r 

gles. 



* These propositions were first given by Thacker in his Mathematical Miscel- 
lany, published in 1743 ; their practical utility has been recently shown by Pro- 
fessor Wallace, in the Edinburgh Philosophical Transactions. 



PLANE TRIGONOMETRY. 139 

19. As the sum of the tangents of any two unequal angles is 
to their difference, so is the sine of the sum of those angles to 
the sine of their difference. 

20. As the sine of the difference of any two unequal angles 
is to the difference of their sines, so is the sum of those sines to 
the sine of the sum of the angles. 

These and other propositions are the foundation of various for- 
mulae, for which the reader who wishes to pursue the inquiry 
may consult the best treatises on Trigonometry. 



III. — Solution of the Cases of Plane Triangles. 

Although the three sides and three angles of a plane triangle, 
when combined three and three, constitute twenty varieties, yet 
they furnish only three distinct cases in which separate rules 
are required. 

Case I. 

When a side and an angle are two of the given parts. 
The solution may be effected by prop. 14 of the preceding 
section, wherein it is affirmed that the sides of plane triangles 
are respectively proportional to the sines of their opposite an- 
gles. 

In practice, if a side be required, begin the proportion with 
a sine, and say, 

As the sine of the given angle, 

To its opposite side ; 
So is the sine of either of the other angles, 
To its opposite side. 
If an angle be required, begin the proportion with a side, 
and say, 

As one of the given sides, 

Is to the sine of its opposite angle ; 
So is the other given side, 

To the sine of its opposite angle. 
The third angle becomes known by taking the sum of the two 
former from 180°. 

Note 1. — Since sines are lines, there can be no impropriety in 
comparing them with the sides of triangles; and the rule is bet- 
ter remembered by young mathematicians than when the sines 
and sides are compared each to each. 



140 PLANE TRIGONOMETRY. 

Note 2. — It is usually, though not always, best to work the 
proportions in trigonometry by means of the logarithms, taking 
the logarithm of the Jirst term from the sum of the logarithms 
of the second and third, to obtain the logarithm of the fourth 
term. Or, adding the arithmetical complement of the loga- 
rithm of the first term to the logarithms of the other two, to 
obtain that of the fourth. 



Case II. 



When two sides and the included angle are given, 

The solution may be effected by means of props. 15 and 16 

of the preceding section. 

Thus: take the given angle from 180°, the remainder will be 

the sum of the other two angles. 

Then say, — As the sum of the given sides, 
Is to their difference ; 
So is the tangent of half the sum of 
the remaining angles, 
To the tangent of half their difference. 
Then, secondly, say, — As the cosine of half the said difference, 
Is to the cosine of half the sum of the angles ; 
So is the sum of the given sides 
To the third, or required side. 
Or, As the sine of half the diff. of the angles, 
Is to the sine of half their sum ; 
So is the difference of the given sides, 
To the third side. 



Example. — In the triangle abc are given 
ac = 450, bc = 540, and the included angle 
c =80°; to find the third side, and the two re- 
maining angles. 



Here bc + ac = 990, bc — A c=90, 180° — c=100° 
= a+b. 

Hence, b c + a c 990 . . Log. = 2*9956352 

To bc— AC 90 . . Log. = 1-9542425 

So is tan \ (a + b) 50° . . Log. = 10-0761865 

So tan h (A— b) 6° 11' Log. = 9-0347938 




PLANE TRIGONOMETRY. 141 

Cos* (a — b) 6° 11' Log.=9-9974660 

Cos h (a + b) 50° Log.=9-8080675 

So is b c+A c 990 Log.=2*9956352 

To A b 640-08 Log.=2'8062367 

Also h (a + b) + i (a— b) = 56° 11' =A ; and h (a + b) 
— h (a — b) = 43°49' =--b. 

Here, much time will be saved in the work by taking 
cos \ (a+b) from the tables, at the same time with tan \ (a+b) ; 
and cos \ (a — b) as soon as tan § (a — b) is found. Observe, 
also, that the log. of b c+a c is the same in the second opera- 
tion as in the first. Thus the tables need only be opened in 
Jive places for both operations. 

Another Solution to Case II. 

Supposing c to be the given angle, and c a, c b, the given 
sides ; then the third side may be found by this theorem, 
viz. 

A B= \/(a c 3 +b c 3 — 2ac.cb.cosc). 
Thus, taking a c=450, b c=540, c=80°, its cos -1736482 
A b= ^/(450 3 +540 3 — 2 . 450. 540 X -1736482) 
= ^[90 3 (5^+^ — 2 . 5. 6 X -1736482)] 
= 90 >/50-58118=90x 7-112=640-08, as before. 



Case III. 

10. When the three sides of a plane triangle are given, to 
find the angles. 

1st Method. — Assume the longest of the three sides as base, 
then say, conformably with prop. 16, 
As the base, 

To the sum of the two other sides ; 
So is the difference of those sides, 

To the difference of the segments of the base. 
Half the base added to the said difference gives the greater 
segment, and made less by it gives the less ; and thus, by 
means of the perpendicular from the vertical angle, divides the 
original triangle into two, each of which falls under the first 
case. 

2d Method. — Find any one of the angles by means of prop. 
18 of the preceding section ; and the remaining angles either 
by a repetition of the same rule, or by the relation of sides to 
the sines of their opposite angles. 
20 



142 PLANE TRIGONOMETRY. 

mi ACHBC 3 A B s AB 3 +BC a — AC 3 

Thus, cos c= — - ; cos b=- 



and cos a = 



2ac.bc 2ab.bc 

B A 2 + A C 2 — B C 3 
2 A B . AC 



Right-angled Plane Triangles. 

1. Right-angled triangles may, as well as others, be solved by- 
means of the rule to the respective case under which any spe- 
cified example falls : and it will then be found, since a right an- 
gle is always one of the data, that the rule usually becomes 
simplified in its application. 

2. When two of the sides are given, the third may be 
found by means of the property in Plane Geom. Triangles, 
prop. 16. 

Hypoth. = ^(base 2 + perp. 3 ) 
Base= v/(hyp. 3 — perp. 2 )= ^/(hyp. + perp.) . (hyp. — perp.) 
Perp. = -v/(hyp. 2 — base 2 ) = v/(hyp.-fbase) . . (hyp. — base.) 

3. There is another method for right angled triangles, known 
by the phrase making any side radius ; which is this. 

"To find a side. — Call any one of the sides radius, and write 
upon it the word radius ; observe whether the other sides be- 
come sines, tangents, or secants, and write those words upon 
them accordingly. Call the word written upon each side the 
name of each side ; then say, 

As the name of the given side, 

Is to the given side ; 
So is the name of the required side, 
To the required side." 

" To find an angle. — Call either of the given sides radius, 
and write upon it the word radius ; observe whether the other 
sides becomes sines, tangents, or secants, and write those words 
on them accordingly. Call the word written upon each side the 
name of that side. Then say, 

As the side made radius, 

Is to radius ; 
So is the other given side, 
To the name of that side, 
which determines the opposite angle." 

4. When the numbers which measure the sides of the tri- 
angle are either under 12, or resolvable into factors which are 
each less than 12, the solution may be obtained, conformably 



HEIGHTS AND DISTANCES. 143 

with this rule, easier without logarithms than with them. 
For, 

Let a b c be a right angled triangle, in which A b, the base, 
is assumed to be radius ; b c is the tangent 
of a, and a c its secant, to that radius ; or di- 
viding each of these by the base, we shall 
have the tangent and secant of a, respec- 
tively, to radius 1. Tracing in like manner 
the consequences of assuming b c, and a c, 
each for radius, we shall readily obtain these expressions. 

j perp. =tan a j e at k ase> 4 _y£: = sec angle at vertex. 

base ° perp. ° 

2. -^-=tan angle at vertex. 5.?^= sin angle at base. 

perp. ° hyp. ° 

3.r^-'=sec angle at base. 6.r^-=sin angle at vertex. 




Section II. — On the Heights and Distances of Objects. 

The instruments employed to measure angles are quadrants, 
sextants, theodolites, &c, the use of either of which may be 
sooner learnt from an examination of the instruments them- 
selves than from any description independently of them. For 
military men and for civil engineers, a good pocket sextant, 
and an accurate micrometer (such as Cavallo's) attached to a 
telescope, are highly useful. For measuring small distances, 
as bases, 50 feet and 100 feet chains, and a portable box of gra- 
duated tape will be necessary. 

We shall here present a selection of such examples as are 
most likely to occur. 

Example I. 

In order to find the distance between two trees a and b, 
which could not be directly measured because of a pool 
which occupied much of the intermediate space, I measured 
the distance of each of them from a third object c, viz. ac = 588, 
b c = 672, and then at the point c took the angle acb between 
the two trees=55° 40'. Required their distance. 

This is an example to case 2 of plane triangles, in which two 
sides, and the included angle, are given. The work, therefore, 
may exercise the student : the answer is 593*8. 



144 HEIGHTS AND DISTANCES. 



Example II. 



Wanting to know the distance between two inaccessible ob- 
jects, which lay in a direct line from the bottom of a tower on 
whose top I stood, I took the angles of depression of the two 
objects, viz. of the most remote 25h°, of the nearest 57°. What 
is the distance between them, the height of the tower being 120 
feet? 

The figure being constructed, as in the 
margin, a b=120 feet, the altitude of the 
tower, and a h the horizontal line drawn 
through its top ; there are given, 

o £ 

h a d=25° 30', hence b a d=b ah — ha d=64° 30'. 

h a c=57° 0', hence b a c =b a h — h a c=33° 0'. 

Hence the following calculation, by means of the natural tan- 
gents. For, if a b be regarded as radius, b d and b c will be the 
tangents of the respective angles b a d, b a c, and c d the dif- 
ference of those tangents. It is, therefore, equal to the product 
of the difference of the natural tangents of those angles into the 
height a b. 

Thus, nat. tan 64£°=2'0965436 
nat. tan 33° =0*6494076 



difference 1-4471360 

multiplied by height 120 



gives distance c d . . 173*6563200 



Example III. 

Standing at a measurable distance on a horizontal plane, 
from the bottom of a tower, I took the angle of elevation of the 
top ; it is required from thence to determine the height of the 
tower. 

In this case there would be given a b and the angle a (see the 
figure in Right-angled Triangles), to find^B c=a b xtan a. 

By logarithms, when the numbers are large, it will be, log. 
b c=log. a B + log. tan A. 



HEIGHTS AND DISTANCES. 145 

Note. — If angle a=11°19' then b c = \ a b very nearly. 



a=16 42 
A=21 48 
A=26 34 
A=30 58 
A=35 
A=38 40 
A=45° 



B C = T q A B 

B C= | A B .... 

B C= | A B .... 

B C= | AB .... 
B C =To A B 

B C= y A B • • • • 

b c= a b, exactly. 



To save the time of computation, therefore, the observer may 
set the instrument to one of these angles, and advance or re- 
cede, till it accords with the angle of elevation of the object ; 
its height above the horizontal level of the observer's eye will 
at once be known, by taking the appropriate fraction of the 
distance A b. 



Example IV. 

Wanting to know the height of a church steeple, to the bot- 
tom of which I could not measure on account of a high wall be- 
tween me and the church, I fixed upon two stations at the dis- 
tance of 93 feet from each other, on a horizontal line from the 
bottom of the steeple, and at each of them took the angle of 
elevation of the top of the steeple, that is, at the nearest station 
55° 54', at the other 33° 20'. Required the height of the 
steeple. 

Recurring to the figure of Example II., we have given the 
distance c d, and the angles of elevation at c and d. The 
quickest operation is by means of the natural tangents, and the 

c d 

theorem ab = — - — . 

cot d — cot c 

Thus cot D=cot 33°20'=1'5204261 

cot c=cot 55 54= -6770509 



Their difference = -8433752 



93 
Hence, a b= nA nn „ r .- =110-27 feet. 
y -8433752 



Example V. 

Wishing to know the height of an obelisk standing at the top 
of a regularly sloping hill, I first measured from its bottom a 
distance of 36 feet, and there found the angle formed by 



146 HEIGHTS AND DISTANCES. 

the inclined plane and a line from the centre of the instrument 
to the top of the obelisk 41° ; but after measuring on downward 
in the same sloping direction 54 feet farther, I found the angle 
formed in like manner to be only 23° 45'. What was the height 
of the obelisk, and what the angle made by the sloping ground 
with the horizon ? 

The figure being constructed as in the margin, there are 
given in the triangle a c b, all the angles and the side a b, to 
find b c. It will be obtained by this pro- 
portion, as sin c (=17° 15'=b — a) : a b 
(=54) :: sin a (=23° 45') : b c=73«3392. J£ 

Then, in the triangle d b c are known b c //im 

as above, b d = 36, c b d = 41° ; to find -• — T^/pHfe 
the other angles, and the side c d. Thus, ^ggfllllllpll 

first, as cb + bdicb- — bd:: tan \ a e 

(d + c)=£ (139°) : tan \ (d— c)=42° 24|'. 

Hence 69° 30' + 42° 24£' = 112° 54§' = c D b, and 69° 30'—- 
42° 24^ '=26° 5i-BCD. Then, sin b c d : b d : : sin c b d : c d 
= 51-86, height of the obelisk. 

The angle of inclination d ae = hda=cdb — 90° = 
22° 54§'. 

Remark. — If the line b d cannot be measured, then the angle 
d a e of the sloping ground must be taken, as well as the angles 
cab and c b d. In that case d a e + 90° will be equal to c d b : 
so that, after c b is found from the triangle a c b, c d may be 
found in the triangle c b d, by means of the relation between 
sides and the sines of their opposite angles. 



Example VI. 

Being on a horizontal plane, and wanting to ascertain the 
height of a tower standing on the top of an inaccessible hill, I 
took the angle of elevation of the top of the hill 40°, and of 
the top of the tower 51°, then measuring in a direct line 180 
feet farther from the hill, I took in the same vertical plane the 
angle of elevation of the top of the tower 33° 45'. Required 
from hence the height of the tower. 

The figure being constructed, as in the mar- 
gin, there are given ab=180, c ab=33° 45', 
acb = c b e — c a e = 17° 15', c b d = 
11°,bdc=180°— (90° — dbe) = 130°. And 
c d may be found from the expression c d 
rad 4 = ab sin a sin c b d cosec a c b sec d b e. 





HEIGHTS AND DISTANCES. 147 

Or, using the logarithms, it will be log. AB+log sin A+log 
sin B+log cosec a c B+log sec dbe — 40 (in the index) = log 
cd ; in the case proposed =log. of 83*9983 feet. 



Example VII. 

In order to determine the distance between two inaccessible 
objects e and w on a horizontal plane, we measured a convenient 
base a b of 536 yards, and at the extremities a and b took the fol- 
lowing angles, viz. b a w=40° 16', w a e=57° 40', a b e=42° 
22', e b w=71° 7'. Required the distance e w. 

First, in the triangle a b e are given all the 
angles, and the side a b to find b e. So, again, 
in the triangle a b w, are all given the angles, and 
a b to find b w. Lastly, in the triangle b e w 
are given the two sides e b, b w, and the in- 
cluded angle e b w to find e w= 939*52 yards. 

Remark. — In like manner the distances taken two and two, 
between any number of remote objects posited round a conve- 
nient station line, may be ascertained. 



Example VIII. 

Suppose that in carrying on an extensive survey, the distance 
between two spires a and b has been found equal to 6594 yards, 
and that c and d are two eminences conveniently 
situated for extending the triangles, but not ad- a b 
mitting of the determination of their distance \\7l — * 
by actual admeasurement : to ascertain it, there- «V~-H? > 
fore, we took at c and d the following angles, viz. \ \{ 
Cacb=85°46' Cadc=31°48' \/ 

\b c d=23° 56' \k d b=68° 2' Y~ „ 

Required c d from these data. 

In order to solve this problem, construct a similar quadrilate- 
ral a c d b, assuming c d equal to 1, 10, or any other convenient 
number: compute a b from the given angles, according to the 
method of the preceding example. Then, since the quadrilate- 
rals a c d b, a c d b, are similar, it will be,as a b :c^::ab:cd; 
and c d is found = 4694 yards. 



148 HEIGHTS AND DISTANCES. 



Example IX. 




Given the angles of elevation of any distant object, taken at 
three places in a horizontal right line, which does not pass 
through the point directly below the object ; and the respective 
distances between the stations ; to find the height of the object, 
and its distance from either station. 

Let a e c be the horizontal plane, f e the 
perpendicular height of the object above 
that plane, a, b, c, the three places of obser- 
vation, f a e, f b e, p c e, the angles of ele- 
vation, and a b, b c, the given distances. 
Then, since the triangles a e f, b e f, c e p, 
are all right angled at e, the distances a e, b e, c e, will mani- 
festly be as the cotangents of the angles of elevation at a, 
b, and c. 

Put a b=d, b c=d, e f = x, and then express algebraically 
the theorem given in Geom. Triangles, 25, which in this case 
becomes, 

AE 3 .BC-f-CE 2 .AB=BE 3 .AC-fAC.AB.BC. 

The resulting equation is 

dx 2 cot 3 a+d x* cot 3 c=(D+fl?) ar 5 cot 2 b + (d + c?) d d. 

From which is readily found 

(D-ffi?) DC? 



x= 4r 



cot 2 a+d cot a c — (v + d) cot 3 b' 

Thus e f becoming known, the distances a e, b e, c e, are 
found, by multiplying the cotangents of a, b, and c, respective- 
ly, by e f. 

Remark. — When d=c?, or v+d=2 d=2 d, that is, when the 
point b is midway between a and c, the algebraic expression 
becomes, 

x—d-k-s/{\ cot 3 A + 5 cot 3 c — cot 3 b), 
which is tolerably well suited for logarithmic computation. 
The rule may, in that case, be thus expressed. 

Double the log. cotangents of the angles of elevation of the 
extreme stations, find the natural numbers answering thereto, 
and take half their sum ; from which subtract the natural num- 
ber answering to twice the log. cotangent of the middle angle 
of elevation : then half the log. of this remainder subtracted 
from the log. of the measure distanced between the first and 
second, or the second and third stations, will be the log. of the 
height of the object. 

The distance from either station will be found as above. 



HEIGHTS AND DISTANCES. 149 

Note. — The case explained in this example, is one that is 
highly useful, and of frequent occurrence. An analogous one 
is when the angles of elevation of a remote object are taken 
from the three angles of a triangle on a horizontal plane, the 
sides of that triangle being known, or measurable : but the 
above admits of a simpler computation, and may usually be 
employed. 



Example X. 

From a convenient station p, where could be seen three ob- 
jects, a, b, and c, whose distances from each other were known 
(viz. a b = 800, ac = 600, b c = 400 yards), I took the hori- 
zontal angles apc = 33° 45', bpc = 22° 30'. It is hence re- 
quired to determine the respective distances of my station from 
each object. 

Here it will be necessary, as preparatory to the computation, 
to describe the manner of 

Construction. — Draw the given triangle abc from any con- 
venient scale. From the point a draw a line 
A d to make with a b an angle equal to 22° 
30', and from b a line b d to make an angle 
dba = 33° 45'. Let a circle be described 
to pass through their intersection d, and 
through the points a and b. Through c and 
d draw a right line to meet the circle again 
in p : so shall p be the point required. For, drawing p a, 
p b, the angle a p d is evidently = a b d, since it stands on 
the same arc a d : and for a like reason b p d = b a d. 
So that p is the point where the angles have the assigned 
value. 

The result of a careful construction of this kind, upon a good 
sized scale, will give the values of p A, p c, p b, true to within 
the 200dth part of each. 

Manner of Computation. — In the triangle abc, where the 
sides are known, find the angles. In the triangle A b d, where 
all the angles are known, and the sides a b, find one of the 
other sides a d. Take bad from b a c, the remainder, d a c 
is the angle included between two known sides, a d, a c ; from 
which the angles a d c and a c d may be found, by chap. iii. 
case 2. The angles c ap = 180° — (apc+acd). Also, 
bcp =«bca — acd: and pbc = abc + p ba = abc + 
sup. adc. Hence, the three required distances are found by 
these proportions. As sin a p c : a c : : sin p a c : p c, and : : sin 
21 o2 




I 50 HEIGHTS AND DISTANCES. 

p c A : p A ; and lastly, as sin b p c : b c : : sin b p c : b p. The 
results of the computation are, pa= 709*33, pc= 1042*66, 
p b = 934 yards. 

\* The computation of problems of this kind, however, 
may be a little shortened by means of an analytical investi- 
gation. Those who wish to pursue this department of trigo- 
nometry may consult the treatises by Bonny castle, Gregory, 
and Woodhouse. 

Note. — If c had been nearer to p than a b, the general prin- 
ciples of construction and computation would be the same ; and 
the modification in the* process very obvious. 



II. Determination of Heights and Distances by approxi- 
mate Mechanical Methods* 

1. For Heights. 

1. By shadows, when the sun shines. — Set up vertically a 
staff of known length, and measure the length of its shadow upon 
a horizontal or other plane ; measure also the length of the 
shadow of the object whose height is required. Then it will 
be, as the length of the shadow of the staff, is to the length of 
the staff itself ; so is the length of the shadow of the object, to 
the object's height. 

2. By two rods or staves set up vertically. — Let two staves, 
one, say, of 6 feet, the other of 4 feet long, be placed upon 
horizontal circular or square feet, on which each may stand 
steadily. Let A b be the object, as a * .. 
tower or steeple, whose altitude is 
required, and a c the horizontal 
plane passing through its base. Let 
c d and e p, the two rods, be placed 
with their bases in one and the same 
line c a, passing through a the foot 
of the object ; and let them be moved 
nearer to, or farther from, each other, until the summit b of the 
object is seen, in the same line as d and f, the tops of the rods. 




HEIGHTS AND DISTANCES. 151 

Then by the principle of similar triangles, it will be, as d h 
(= c e) : f h : : d g (= c a) : b g ; to which add a g = c d, 
for the whole height a b. 

3. By Reflection. — Place a vessel of water upon the ground, 
and recede from it, until you see the top of the object reflected 
from the smooth surface of the liquid. Then, since by a prin- 
ciple in optics, the angles of incidence and reflection are equal, 
it will be as your distance measured horizontally from the point 
at which the reflection is made, is to the height of your eye 
above the reflecting surface ; so is the horizontal distance of the 
foot of the object from the vessel to its altitude above the said 
surface.* 

4. By means of a portable barometer and thermometer. — 
Observe the altitude b of the mercurial column, in inches, tenths, 
and hundredths, at the bottom of the hill, or other object whose 
altitude is required ; observe, also, the altitude, b, of the mercu- 
rial column at the top of the object ; observe the tempera- 
tures on Fahrenheit's thermometer, at the times of the two 
barometrical observations, and take the mean between them. 

B 

Then 55000 x 7 = height of the hill, in feet, for the tem- 

b + 

perature of 55° on Fahrenheit. Add T ^_ of this result for 
every degree which the mean temperature exceeds 55° ; subtract 
as much for every degree below 55°. 

This will be a good approximation when the height of the 
hill is below 2000 ; and it is easily remembered, because 55°, 
the assumed temperature, agree with 55, the effective figures in 
the coefficient ; while the effective figures in the denominator 
of the correcting fraction are two fours. 

*' m * Where great accuracy is required logarithmic rules be- 
come necessary, of which various are exhibited in treatises on 
Pneumatics. The following, by the Rev. W. Galbraith, of 
Edinburgh, is a very excellent approximation. 

For Fahrenheit's thermometer. 

b — b ^ _ , t+t' 
b + b 

+ h (0.00268 + 0.00268 cos 2 x + 0.00000005 h) 
in which h is the true height in feet, t the temperature of the 



h= $48400 + 60 (*+/')£- tS— $2.42 + — ?(t— t'] 

C v y 5 B + 6 I ^ 300 3 v ; 



* Leonard Digges, in his curious work the Pantometria, published in 1571, 
first proposed a method for the determination of altitudes by means of a geometri- 
cal square and plummet, which has been described by various later authors, as 
Ozanam, Donn, Hutton, &c. But as it does not seem preferable to the methods 
above given, I have not repeated it here. 



152 HEIGHTS AND DISTANCES. 

air by detached thermometer at the lower station, t that at 
the upper ; T the temperature of the mercury in the barome- 
ter at the lower station by the attached thermometer, t 1 that 
at the upper ; b the height of the mercury in the barometer at 
the lower station, b that at the upper, h the height, and *■ the 
latitude. 

5. By an extension of the principle of pa. 145. — Set the 
sextant, or other instrument, to the angle of 45°, and find the 
point c (pa. 144.) on the horizontal plane, where the object a b has 
that elevation : then set the instrument to 26° 34', and recede 
from c, in direction b c d, till the object has that elevation. 
The distance c d between the two stations ivill be — a b 

So, again, if c = 40°, d = 24° 31^', c d will be = a b. 
or, if c == 35°, d = 22° 23', c d = a b. 

or, if c = 20°, d = 20° 6', c d = a b. 

or, if c = 20°, d = 14° 56', c d =ab. 

or, generally, if cot d — cot c = rad. c d = a b. 

6. For deviation from level. — LetE represent the elevation 
of the tangent line to the earth above the true level, in feet and 
parts of a foot, d the distance in miles : then e = | d 2 . 

This gives 8 inches for a distance of one mile ; and is a 
near approximation when the distance does not exceed 2 or 3 
miles. 



2. For Distances. 

1. By means of a rhombus set off upon a horizontal plane. 

Suppose o the object and o b the required distance. With 
a line or measuring tape, whose length is equal to the side 
of the intended rhombus, say 50 or 100 feet, lay down one 
side b a in the direction b o towards the object, 
and b c another side in any convenient direction 
(for whether b be a right angle, or not, is of no 
consequence) ; and put up rods or arrows at a 
and c. Then fasten two ends of two such lines 
at a and c, and extend them until the two other 
ends just meet together at d ; let them lie thus 
stretched upon the ground, and they will form b 

the two other sides of the rhombus a d, c d. Fix a mark or 
arrow at r, directly between c and o, upon the line a d ; 
and measure r d, r a upon the tape. Then it will be as 
R d : d c : : c b : b o, the required distance. 




HEIGHTS AND DISTANCES. 153 

Otherwise. To find the length of the inaccessible line q r. 

At some convenient point b, lay down the rhombus 
b A d c, so that two of its sides 
b A, b c, are directed to the ex- 
tremities of the line q r. Mark 
the intersections o and p, of 
a r, c q, with the sides of the 
rhombus, (as in the former me- 
thod) : then the triangle odp 
will be similar to the triangle 
r b q ; and the inaccessible dis- 
tance r q will be found 




o d X d p 
Thus, if b a = b c, &c. = 100 f. o d = 9 f. 5 in., dp = 

11 f. 10 in. o p = 13 f. 7 in. then q r = l ™° X |fff = 

y TT X X1 T¥ 

1219 feet. 

2. By means of a micrometer attached to a telescope. 

Portable instruments for the purpose of measuring extremely 
small angles, have been invented by Martin, Cavallo, Dollond, 
Brewster, and others. In employing them for the determina- 
tion of distances, all that is necessary in the practice is to 
measure the angle subtended by an object of known dimen- 
sions, placed either vertically or horizontally, at the remoter 
extremity of the line whose length we wish to ascertain. 
Thus, if there be a house, or other erection, built with bricks, 
of the usual size ; then four courses in height are equal to a 
foot, and four in length equal to a yard: and distances measured 
by means of these will be tolerably accurate, if care be taken 
with regard to the angle subtended by the horizontal object, to 
stand directly in front of it. A man, a carriage wheel, a win- 
dow, a door, &c. at the remoter extremity of the distance we wish 
to ascertain, may serve for an approximation. But in all cases 
where it is possible, let a foot, a yard, or a six-feet measure, be 
placed vertically, at one end of the line to be measured, while 
the observer with his micrometer stands at the other. Then 
if h be the height of the object, 

either \ h x cot \ angle subtended 
or h x cot angle 

A B 2 

* For fs:di::ab:is= ; 

p D 

A B 2 .0 P 

and od:op;:br!R« = . 



154 HEIGHTS AND DISTANQES. 

will give the distance, according as the eye of the observer id 
horizontally opposite to the middle, or to one extremity of the 
object whose angle is taken. 

When a table of natural tangents is not at hand, a very near 
approximation for all angles less than half a degree, and a tole- 
rably near one up to angles of a degree, will be furnished by the 
following rules. 

1. If the distant object whose angle is taken be 1 foot in length, 
then 

3437*73 -J- the angle in minutes > will give the distance 
or 206264*-r-the angle in seconds } in feet. 

2. If the remote object be 3, 6, 9, &c. feet in length, multiply 
the former result by 3, 6, 9, &c. respectively. 

Ex. 1. What is the distance of a man 6 feet high, when he 
subtends an angle of 30 seconds ? 

206264 x6-r30 = 206264 -f- 5 =41252*8 feet= 13750*9 
yards, the distance required. 

Ex. 2. In order to ascertain the length of a street, I put up 
a foot measure at one end of it, and standing at the other found 
that measure to subtend an angle of 2 minutes : required the 
length of the street. 

3437.73 -j- 2 = 1718*86 feet = 572*95 yards. 



3. By means of the velocity of sound. 

Let a gun be fired at the remoter extremity of the required 
distance, and observe by means of a chronometer that measures 
tenths of seconds, the interval that elapses between the flash and 
the report : then estimate the distance for one second by 
the following rule, and multiply that distance by the observed 
interval of time ; the product will give the whole distance 
required. 

At the temperature of freezing, 33°, the velocity of sound is 
1100 feet per second. 

For lower temperatures deduct > , ir r 
For higher temperatures add 5 ! 

t th 1 1 on \ ^ or ever y degree °f difference from 33° on 
Fahr. therm. ; the result will show the velocity of sound, very 
nearly, at all such temperatures. 

Thus, at the temperature of 50°, the velocity of sound is, 

1100 xh (50 — 33) = 1108£ feet. 
At temperature 60°, it is 1100 + * (60 — 33) =1113$ feet. 



CONIC SECTIONS. 155 



CHAPTER V. 
CONIC SECTIONS. 

1. Conic Sections are the figures made by a plane cutting a 
cone. 

2. According to the different positions of the cutting plane 
there arise five different figures or sections, viz. a triangle, a 
circle, an ellipsis, an hyperbola, and a parabola : of which the 
three last are peculiarly called Conic Sections, 

3. If the cutting plane pass through the vertex of the cone, 
and any part of the base, the section will be a triangle. 

4. If the plane cut the cone parallel to the base, or make no 
angle with it, the section will be a circle. 

5. The section is an ellipse when the cone is cut obliquely 
through both sides, or when the plane is inclined to the base in 
a less angle than the side of the cone is. 

6. The section is a parabola, when the cone is cut by a plane 
parallel to the side, or when the cutting plane and the side of the 
cone make equal angles with the base. 

7. The section is an hyperbola, when the cutting plane 
makes a greater angle with the base than the side of the cone 
makes. 

8. And if all the sides of the cone be continued through 
the vertex, forming an opposite equal cone, and the plane be 
also continued to cut the opposite cone, this latter section will 
be the opposite hyperbola to the former. 

9. The vertices of any section are the points where the 
cutting plane meets the opposite sides of the cone, or the 
sides of the vertical triangular section ; as A and b, in the figs, 
below. 

Hence the ellipse and the opposite hyperbolas have each two 
vertices ; but the parabola only one ; unless we consider the 
other as at an infinite distance. 

10. The major axis, or transverse diameter, of a conic section, 
is the line or distance a b between the vertices. 

Hence the axis of a parabola is infinite in length, a b being 
only a part of it. 



156 



CONIC SECTIONS. 



Ellipse. 



Hyperbolas. 



Parabolas. 





11. The centre c is the middle of the axis. 

Hence the centre of a parabola is infinitely distant from the 
vertex. And of an ellipse, the axis and centre lie within the 
curve : but of an hyperbola, without. 

12. A diameter is any right line, as a b or d e, drawn 
through the centre, and terminated on each side by the curve ; 
and the extremities of the diameter, or its intersections with the 
curve, are its vertices. 

Hence all the diameters of a parabola are parallel to the 
axis, and infinite in length. Hence also every diameter of 
the ellipse and hyperbola has two vertices ; but of the para- 
bola only one ; unless we consider the other as at an infinite 
distance. 

13. The conjugate to any diameter is the line drawn through 
the centre, and parallel to the tangent of the curve at the vertex 
of the diameter. So, f g, parallel to the tangent at d, is the 
conjugate to d e ; and h i, parallel to the tangent at a, is the 
conjugate to a b. 

Hence the conjugate h i, of the axis a b, is perpendicular to 
it, and is often called the minor axis. 

14. An ordinate to any diameter is a line parallel to its 
conjugate, or to the tangent at its vertex, and terminated by 
the diameter and curve. So d k and e l are ordinates to 
the axis a b ; and m n and n o ordinates to the diameter 
d e. — Hence the ordinates of the axes are perpendicular to 
it ; but of other diameters, the ordinates are oblique to 
them. 

15. An absciss is a part of any diameter, contained between 
its vertex and an ordinate to it ; as a k or b k, and d n or e n. 
Hence, in the ellipse and hyperbola, every ordinate has two ab- 
scisses ; but in the parabola only one ; the other vertex of the 
diameter being infinitely distant. 

16. The parameter of any diameter is a third proportional 
to that diameter and its conjugate. 

17. The focus is the point in the axis where the ordinate is 
equal to half the parameter : as k and l, where d k or e l is 



CONIC SECTIONS. 



157 



equal to the semiparameter. — Hence the ellipse and hyperbola 
have each two foci, but the parabola only one. The foci, or 
burning points, were so called, because all rays are united or re- 
flected into one of them, which proceed from the other focus, 
and are reflected from the curve. 

18. The directrix is a line drawn perpendicular to the axis 
of a conic section, through an 
assignable point in the prolon- 
gation of that axis ; such that 
lines drawn from that directrix 
parallel to the axis to meet the 
curve, shall be to lines drawn 
from the points of intersection 
to the focus, in a constant ratio 
for the same curve. Thus, if e m : M r : : E r m' : m' p, then 
t e ' e is the directrix. The curve will be a parabola, an ellipse, 
or an hyperbola, according as f m is equal to, less than, or 
greater than, m e. 

19. An asymptote is a right line towards which a certain 
curve line approaches continually nearer and nearer, yet so as 
never to meet, except both be produced indefinitely. The 
hyperbola has two asymptotes. 




Section I. — Properties of the Ellipse. 



1. If in the annexed diagram the 
ellipse b k d n, be cut from the frustrum 
of the right cone, the diameter of whose 
ends are e d, b c. Then, if b d be the 
transverse, or major axis, k n the con- 
jugate, or minor axis, and s, s', the foci, 
we shall have 



E D 


£Or |\ 



W C 



B D 2 = D C a + E D 


B C . . 


. . (l) 


K N 2 = E D . B C . 


. . . 


• • (2) 


SS' = EB = DC . 




• • (3) 



2. If A b, a b, be the two axes, c the_ 
centre, r, /, the foci, p any point in the 
curve, p d, an ordinate : also, if a b — 
2 t, ab~2cGi> — x\ KB — X, DP = 
y, f p = z, angle p f d = <j>, y/ (t 2 — c 3 ) 
22 P 



' 1 

A '■ / — 


^~ 


a 




^ 


) 


C 


f ) 



158 



CONIC SECTIONS : ELLIPSE. 



y*=j>{t*-x'*) . 



(I) 
(II) 



z = 



t — d cos <j> 



(III) 



The first of these is the equation of the curve when the 
abscissae are reckoned from the extremity a of the transverse 
axis : the second is the equation when the abscissae are 
reckoned from the centre c : and the third is called the polar 
equation, and is principally used in the investigations of 
astronomy. 

Ex. Suppose a b = 20, a b = 12, a d =4. Reouired the 
numeral value of p d. 

c 3 36 / 

Here y* = -(2 tx — x*) = — ((20 x 4) — 16) 

36 _ 6 2 . 8 3 

~~ Too * 64 ~" 10 a ' 



Consequently y = 



6x8 
10 



4-8 



P D. 



Or, taking Equa. II. where c d = x' = 10 — 4 = 6. and t 
and c as before we have 



36 



6 3 . 8 3 

To 3 " 



y* = _ U* — x' 2 ) = (100 — 36) = — -— , as before. 

* t* ' 100 v ; 

3. In the same figure, we have 



a <r : a (r : : a d . d b : d p 



... (4) 
also p p + p/= a b (5) ; and f/ 3 = a b 3 — a b* .... (6) 

4. Let t k be a tangent to 
the ellipse at any point k, and 
let t be the point where that 
tangent meets the prolonga- 
tion of the axis : let also p h, 
fh be perpendiculas from the 
foci, F,f, upon the tangent, and 
let g h = f h : then 

Z. ppt = Z./pk....(7) c d : c A : : c a : c T (8) 

h and h fall in circumf. of circle whose diam. is A b .... (9) 

If m be in the middle of p d, then a m produced will meet 

the two tangents t k, b k, in their point of intersection k (10.) 




conic sections: ellipse. 159 

If d the foot of the ordinate pass through f the focus, then 
the point, t, of intersection of the tangent and the prolonga- 
tion of the axis will be the point t of the directrix (Def. 
18).... (11.) 

ph./ h=c b* (12) p h 3 =c b* . ~ . . . , (13) 

5. If an ordinate be drawn to any diameter of an ellipse, then 
will the rectangle of the abscissae be to the square of the ordinate 
in a given ratio (14) 

6. All the parallelograms that may be circumscribed about an 
ellipse are equal to one another : and every such parallelogram 
is equal to the rectangle of the two axes (15) 

7. The sum of the squares of every pair of conjugate diame- 
ters is equal to the same constant quantity ; viz. the sum of the 
squares of the two axes (16) 

8. Def — The radius of curvature of a conic section or 
other curve is the radius of that circle which is precisely of the 
same curvature as the curve itself, at any assigned point, or the 
radius of the circle which fits the curve, and coincides with it, 
at a small distance on each side of the point of contact. The cir- 
cle itself is called the osculatory circle, or the equicurve circle ; 
and if the curve be of incessantly varying curvature, each point 
has a distinct equicurve circle, the radius of which is perpen- 
dicular to the tangent at the point of contact. 

9. Let p c be the radius of curvature at any point p in an el- 
lipse or hyperbola whose major axis is a b, minor axis a b, and 

foci, f, /, then is p c = (jLl^Jjl (17) 

9 ■ i a b . a b ' 

The radius of curvature is greatest at the extremities of the 

A B 3 

minor axis, when it is = — ? ..... (18) 
a b v f 

The radius of curvature is least at the extremities of the 

a b 2 
major axis, when it is = (19) 



10. Prob. To construct an ellipse whose two axes are given. 
Find the distance f f, from the value of 
F / 2 , given in equa. 6, or from f /= 
-v/a b 2 — a b 2 . Then, let a fine thread, 
f p / f, in length==F/+A b, be put round 
two pins fixed at the points T,f: then, if 
a pencil be put within the cord, and the 
whole become tightened so as to make three right lines f p, 




160 



CONIC SECTIONS I ELLIPSE. 




p f f f, the point p may be carried on, the cord slipping round 
the fixed pins f, f, so as to describe and complete the ellipse 

APjOB^A. 

Otherwise. — Let there be provided 
three rulers, of which the two r i, /h, 
are of the same length as the transverse 
axis a b, and the third h i, equal in 
length to f f, the focal distance. Then 
connecting these rulers so as to move 
freely about p f and about h i, their 
intersection p will always be in the 
curve of the ellipse : so that, if there 
be slits running along the two rulers, 

and the apparatus turned freely about the foci, a pencil put 
through the slits at their point of intersection will describe the 
curve. 

* # * There are various other methods, as by the elliptic com- 
passes, the trammels, &c. But the first of the above methods is 
as accurate and easy as can well be desired. 

11. Prob. To find the two axes of any proposed ellipse. 
Draw any two parallel lines 

across the ellipse, as m l, f k : bi- 
sect them in the points i and d, 
through which draw the right line 
n i d o, and bisect it in c. From o 
as a centre, with any adequate radius, 
describe an arch of a circle to cut the 
ellipse in the points g, h. Join g, h, 
and parallel to the line g h draw 
through c the minor axis a b ; perpendicular to which through 
c draw A b, it will be the major axis. 

12. Prob. From any given point out of an ellipse to draw 
a tangent to it. 

Let t be the given point, through it and the centre c draw the 
diameter a b ; and parallel to it any line h i terminated by the 
curve. Bisect h i in o : 
and c o produced will be 
the conjugate to a b. — 
Draw any line t s=t b, 
and make t r = t c. 
Draw r a, and parallel 
to it s p cutting a b in p. 
Through p, draw p m pa- 
rallel to c d, and join t m, 
It will be the tangent required. 





CONIC SECTIONS : HYPERBOLA. 



161 



Section II. — Properties of the Hyperbola. 



1. If, in the annexed diagram, the 
conjugate hyperbolas whose vertices 
are d, b, are cut from the two opposite 
right cones whose common summit is 
v, and b c, d e, be the diameters of 
the two circular bases of the two 
cones; then d b, k n, being the 
axes, and s, s', the foci, we shall have 




d b^ = d c 2 

K N 2 

S s' 



DE . B C . . . . (1) 

D E . B C (2) 

— E B = D C (3) 

If these three properties be compared with the corres- 
ponding ones for the ellipse, they will be found to agree, 
with the simple difference of the signs + and — of the con- 
necting quantities in the first property. This at once indi- 
cates a general analogy between the properties of the two 
curves. 

2. Hence, putting ac=cb = £, a c = c b = c, c f = g?, 
a d = x, c d = x', d p = y, angle p f d = <?>, z = f p, we have 




= - (2 t x + x 2 



-=-(*»-«■) 



(I) 
(II) 



(III) 



t + d cos <p 

for the three most useful forms of the equation to the hyper- 
bola, agreeing with those to the hyperbola, except in the 
signs. 

3. And hence it follows, taking this and the preceding 
marginal figures to correspond with 
those in arts. 3 and 4 Ellipse ; that the 
properties indicated by the parenthe- 
tical figures (4), (5), (6), (7), (8), (9), 
(U),(12), (13), (14), (15), and (16), 
hold in the hyperbola ; simply chang- 
ing + to — in (5), — to + in (6), 
circumscribed to inscribed between the four hyperbolas in (15), 
and sum to difference in (16). Those properties, therefore, 
need not be here repeated. 

p 2 




162 



CONIC SECTIONS : HYPERBOLA, 



4. Besides the above, however, there are 
several curious properties which relate to 
the asymptotes of the hyperbola ; some of 
the most useful being these : viz. 
m c, m c, being asymptotes, p d p a double 
ordinate, b h, p f, parallel tocm; &c. 




parallelogram c 



H B K 



parallelogram cfpg 
parallelogram cfp g 



m p. p p = mp .p p = c b 3 (18) ;.. mp = pm . 



(17) 
(19) 



triangle cpt = triangle c b k (20) former diagram. 

5. Also, if the abscissae c d, c e, c f, &c. of any hyperbola, 
be taken on one of the asymptotes in an increasing geometri- 
cal progression, the ordinates d b, e g, 
f h, &c. parallel to the other asymp- 
tote are in decreasing geometrical 
progression, having the same ratio . . 
(21). 

6. And, when the distances c e, c f, / / / 
&c. are in geometrical progression, c d e 
the asymptotic spaces degb,df h b, &c. will be in arithmeti- 
cal progression, and will, therefore, be analogous to the loga- 
rithms of the former. The nature of the system of logarithms 
will depend upon the value of the angle made by the two 
asymptotes. In Napier's logarithms l c f is a right angle : in 
the common logarithms l c f is 25° 44' 21"\. 




7. To describe Hyperbolas. 





Let one end of a longruler/M o be fastened at the point/ 
by a pin on a plane, so as to turn freely about that point as a 
centre. Then take a thread f m o, shorter than the ruler, and 



CONIC SECTIONS I PARABOLA. 163 

fix one end of it in f, and the other to the end o of the ruler. 
Then if the ruler f m o be turned about the fixed point f, at 
the same time keeping the thread o m f always tight, and its 
part m o close to the side of the ruler, by means of the pin m ; 
the curve line A x described by the motion of the pin m is 
one part of an hyperbola. And if the ruler be turned, and 
move on the other side of the fixed point f, the other part 
A z of the same hyperbola may be described after the same 
manner. — But if the end of the ruler be fixed in f, and that 
of the thread in /, the opposite hyperbola x a z may be de- 
scribed. Otherwise : also by continued motion. Let c and 
f be the two foci, and e and k the two vertices of the hyper- 
bola. (See the last fig. above.) Take three rulers, c d, d g, g f, 
so that cd = gf = ek, and dg = cp; the rulers c d and 
G f being of an indefinite length beyond c and g, and having 
slits in them for a pin to move in ; and the rulers having 
holes in them at c and f, to fasten them to the foci c and 
f by means of pins, and at the points d and g they are to 
be joined by the ruler d g. Then, if a pin be put in the 
slits, viz. at the common intersection of the rulers c d and 
g f, and moved along, causing the two rulers g f, c d, to 
turn about the foci c and f, that pin will describe the portion e e 
of an hyperbola. 



Section III. — Properties of the Parabola. 



1. Let the right cone a b c in the mar- 
ginal figure, have a parabolic section, l d g, 
whose focus is f, vertex d, base l g ; from 
d let fall the perpendicular d p upon the 
side a b of the cone ; let e p be bisected in 
s : also, let a plane be cut through s paral- 
lel to b c, and continued to meet the plane 
of the parabola, in r x. 



Then .... r x is the directrix of the parabola . . 



E D /n\ 

d f= (2) 

4ae ' 

DF = i^-.. (3) 

4ed v ; 

2ep=4dp = parameter (4) 




164 



CONIC SECTIONS : PARABOLA. 



2. Let p = parameter of a parabola, x = A p any absciss, 
y = p k, the corresponding ordinate, z — f x, d = a f, <& = 
angle kfp,f being the focus : then 
y*=px (I) 

• (id 



E' E 



E E 



Z = 



1 =F cos <p 
the equations to the parabola : in the 
latter of which, or the polar equation, 
the sign + obtains when p is between 
when p is below f. 



^: 



A and f, and 



Rad. of curvature at x 



(Ix+pY 



(III) 



2 Vp 
at the vertex, a x vanishes, and we have 

rad. of curv. at vertex = § p (IV) 

3. In the same figure, where e" e' g e is the directrix, the 
following properties obtain : viz. 

A F = A G, F D = D E, E K = K E', F M = M e", &C. . (5) 

A P PL 2 



A p : A q :: p is : q n , or = 



(6) 



(9) 



A Q QN' 

where a p and a q are any abscissae, and pl,qn, their corres- 
ponding ordinates. 

FK = AP + AF, FM = AQ + AF (7) 

af=5FH — ^dh and d h = parameter (8) 

As p : qn + p l::qn — pl:aq — a p 
or, as p : m o : : o n : o l 

4. Again, let p t be the tangent to 
a parabola at any point p, and let 
h p d be drawn through p parallel to 
the axis a k ; let p x be perpen- 
dicular to t p : then is g t the subtan- 
gent, p k the normal, g k the subnor- 
mal ; and the following properties ob- 
tain : viz. 



I 


A 


T 

B D 


p 


t} 


A 

F N. 




H > 


K 



angle fpt = angle ptp = angle tpd (10) 

angle k p h = angle kpf (11) 

p p = p T (12) G A = A T . . . (13) 

subtangent gt = 2ag (14) 

subnormal g k=2 a f=| param. a constant auan. (15) 



CONIC SECTIONS 1 PARABOLA. 



165 




5. In the marginal figure also, where 
c q is a tangent to the parabola at the 
point c, and i k, o m, q l, &c. parallel 
to the axis a d. 

Then ie:ek::ck:kl....(16) 
and a similar property obtains, whether 
c l be perpendicular or oblique to t d. 

The external parts of the parallels ie,ta,on,pl, &c. are 
always proportional to the squares of their intercepted parts of 
the tangent ; that is, 

the external parts i e, t a, o n, p l, } 

are proportional to c r, c t 3 , c o 2 , c p 2 , > (17) 

or to the squares c k 2 , c d 2 , c m 2 , c l 2 , ) 

And as this property is common to every position of the 
tangent, if the lines i e, t a, o n, &c. be appended to the points 
i, t, o, &c. of the tangent, and moveable about them, and of such 
lengths that their extremities e, a, n, &c. be in the curve of a 
parabola in any one position of the tangent ; then making the 
tangent revolve about the point c, the extremities e, a, n, &c. 
will always form the curve of some parabola, in every position 
of the tangent. 

The same properties, too, that have been shown of the axis, 
and its abscisses and ordinates,&c. are true of those of any other 
diameter. 




6. Prob. To construct a Parabola. 

Construct an isosceles triangle a b d, whose base a b shall 
be the same as that of the proposed parabola, 
and its altitude c d twice the altitude c v of 
the parabola. Divide each side A d, d b, 
into 10, 12, 16, or 20, equal parts [16 is a good 
number, because it can be obtained by conti- 
nual bisections], and suppose them numbered 
1,2, 3, &c. from a to d, and 1,2, 3, &c. from 
d to b. Then draw right lines 1, 1 ; 2, 2 ; 3, 3 ; 
4, 4 ; &c. and their mutual intersection will 
beautifully approximate to the curve of the parabola avb. 

Otherwise : by continued motion. — Let the ruler, or direc- 
trix b c, be laid upon a plane 
with the square g d o, in such 
manner that one of its sides d g 
lies along the edge of that ruler ; 
and if the thread f m o equal 
in length to d o, the other side 
of the square have one end fixed 
in the extremity of the ruler at 
23 




166 



CONIC SECTIONS ! PARABOLA. 



Bisect d e, h i, in o 




o, and the other end in some point r : then slide the side of 
the square d g along the ruler b c, and at the same time keep 
the thread continually tight by means of the pin m, with its 
part m o close to the side of the square do; so shall the curve 
a m x, which the pin describes by this motion, be one part of a 
parabola. 

And if the square be turned over, and moved on the other 
side of the fixed point p, the other part of the same parabola 
A m z will be described. 

7. Prob. Any right line being given in a parabola, to find 

the corresponding diameter : also, the axis, parameter, and 

focus. 

Draw h i parallel to the given line d e. 
and g, through which draw a o g for 
the diameter. 

Draw h r perp. to a g and bisect 
it in b ; and draw v b parallel to a g, 
for the axis. 

Make v b : h b : : h b : parameter 
to the axis. 

Then i the parameter set from v to f gives the focus. 

8. Prob. To draw a Tangent to a Parabola. 

If the point of contact c be given, draw the ordinate c b, and 
produce the axis until at = a b : then join t c, which will be 
the tangent. 

Or if the point be given in the axis produced : take ab = 
a t, and draw the ordinate b c, which will give c the point of 
contact ; to which draw the line t c as before. 

If d be any other point, neither in the curve nor in the axis 
produced, through which the tan- 
gent is to pass : draw d e g perpen- 
dicular to the axis, and take d h a 
mean proportional between d e and 
d G, and draw h c parallel to the 
axis; so shall c be the point of contact, 
through which and the given point 
d the tangent d c t is to be drawn. 

When the tangent is to make a given angle with the ordi- 
nate at the point of contact : take the absciss a i equal to 
half the parameter, or to double the focal distance, and draw 
the ordinate i e : also draw a h to make with a i the angle 
a h i equal to the given angle ; then draw h c parallel to the 
axis, and it will cut the curve in c the point of contact, where a 
line drawn to make the given angle with c b will be the tangent 
required. 




CONIC SECTIONS I JOINTS OF VOUSSOIRS. 



107 




., 0" 



Section IV. — General Application to Architecture. 

Prob. 1. To find, by construction, the position of the joints 
of the voussoirs, to a parabolic arch. 

In the practice of arcuation, the voussoirs or arch-stones are 
so cut that their joints are perpendicular to the arch or to its 
tangent, at the points where they respect- 
ively fall. Hence, if a v b be the proposed 
parabola, p, p' ? p", &c. the points at which 
the positions of the joints are to be deter- 
mined : draw the ordinates p m, p' m', 
p" m", and on the prolongation of the axis 
set off v t=v m, v t'=v m', v t"=v m", 
&c. Join t p, t' p', t" p", &c. and per- 
pendicular to them respectively the lines 
p o,p' o', p" o", &c. ; they will determine 
the positions of the joints required. 

Prob. 2. To find the same for an elliptical arch. 

Let a b be the span of the arch, and 
a p p' p" b the arch itself, of which f 
and f are the foci. Draw lines p p, 
/p,from the foci to each of the points 
p : bisect the respective angles p p f, 
F p r /, p p" f y by the lines p o, p' o', 
p" o" ; they will show the positions of the joints at the points 
p, p', p". 

Prob. 3. To find the same for a cycloidal arch* 

Let a v b be the cycloid, c p y q its generating circle, and 
p, p', p", points in the 
arch where joints will 
fall. Draw the ordi- 
nates pra, p'ra', p"ra", 
each parallel to the 
base ab of the cycloid, 
and cutting the circle 
in the points p, p', p". Join v p, v p', v p", and perpendicular 
to each the lines p o, p' o',p" o" ; parallel to each of which 
respectively draw po, p' o', p" o" ; they will mark the positions 
of the joints at the several points proposed. 

* This problem is introduced here, as belonging to the subject of arcuatio n 
although it depends upon a property of the cycloid described hereafter, viz. that the 
tangent to any point p of a cycloid is parallel to the corresponding chord vp of the 
generating circle. 





168 



CURVES : CONCHOID. 



CHAPTER VI. 
CURVES, 

*/l knowledge of which is required by Architects and Engineers. 

Section I. — The Conchoid. 



Conchoid, or Conchiles, is the name given to a curve by- 
its inventor, Nicomedes, about 200 years before the Christian 
era. 

The conchoid is thus constructed : a p and b d being two 
lines intersecting at right angles : from p draw a number of 
other lines, pfde, &c. on which take always de = df = ab 
or b c ; so shall the curve line drawn through all the points 
e, e, e, be the first conchoid, or that of Nicomedes ; and the 
curve drawn through all the other points, p, r, p, is called the 
second conchoid ; though, in reality, they are both but parts of 
the same curve, having the same pole p, and four infinite legs, 
to which the line d b d is a common asymptote. 




The inventor, Nicomedes, contrived an instrument for de- 
scribing his conchoid by a mechanical motion : thus, in the 
ruler dd is a channel or groove cut, so that a smooth nail 
firmly fixed in the moveable ruler c a, in the point d, may slide 
freely within it : into the ruler a p is fixed another nail at p, for 
the moveable ruler a p to slide upon. If therefore the ruler a p 
be so moved as that the nail d passes along the canal d d, the 
style, or point in a, will describe the first conchoid. 



CURVES : CISSOID. 



169 



Conchoids of all possible varieties may also be constructed 
with great facility by Mr. Jopling's apparatus for curves, now- 
well known. 

Let a b=b c=d e = d F=a, p B=b, b g=e h=#, and G E 
=BH=y: then the equation to the first conchoid will be 
x 2 (b+x) 2 + x 2 y 2 =a 2 (b-\-x) 2 , or x 4 +2 b x*+b 2 a? + x 2 y 2 
= a 2 b 2 +2 a 2 bx + a 2 x 2 ; and, changing 
only the sign ofx, as being negative in the 
other curve, the equation to the 2d con- 



y 2 = a 2 





choid will be x 2 (b — x) 2 + x 2 

(b __ x y 9 orx 4 — 2bx 3 +b 2 x 2 + x 2 y a 

= a 2 b 2 —2 a 2 bx+a 2 x 2 . 

Of the whole conchoid, expressed by 
these two equations, or rather one equation 
only, with different signs, there are three 
cases or species : as first, when b c is less 
than b p, the conchoid will be as in the 2d 
fig. above ; when b c is equal to b p, the 
conchoid will be as in the 3d fig. ; and 
when b c is greater than b p, the conchoid 
will be as in the 4th or last fig. 

Newton approves of the use of the conchoid for trisecting 
angles, or finding two mean proportionals, or for constructing 
other solid problems. But the principal modern use of this 
curve, and of the apparatus by which it is constructed, is to 
sketch the contour of the section that shall represent the dimi- 
nution of columns in architecture. 

The fixed point p is called the pole of the conchoid ;dddd 
the directrix : it is an asymptote to both the superior and the 
inferior conchoid. In the last figure the inferior conchoid is also 
nodated. 



Section II. — The Cissoid or Cyssoid. 

The cissoid is a curve invented by an ancient Greek geome- 
ter and engineer named Diodes, for the purpose of finding two 
continued mean proportionals between two given lines. This 
curve admits of an easy mechanical construction ; and is de- 
scribed very beautifully by means of Mr. Jopling's apparatus. 

At the extremity b of the diameter a b, of a given circle 
a o b o f erect the indefinite perpendicular e b e, and from the 

Q 



170 



CURVES : CYCLOID. 



other extremity A draw any number of right lines, ac,ad,ae, 
&c. cutting the circle in the points r, o, m, &c. ; then, if c l be 
taken=A r, d o=a o, e n=a m, &c, the curve passing through 
the points, A, l, o, n, &c. will be the cissoid. 




e i c B CD 

1. Here the circle a o b o is called the generating circle ; and 
a b is called the axis of the curves a l o n, &c. a I o n, &c. 
which meet in a cusp at a, and, passing through the middle 
points o, o, of the two semicircles, tend continually towards the 
directrix, e b e, which is their common asymptote. 

2. If a o and a o are quadrants, the curve passes through o 
and o, or it bisects each semicircle. 

3. Letting fall perpendiculars l p, r q, from any correspond- 
ing points l r : then is a p=b q, and a l=c r. 

4. a p : p b : : p l 3 : a p s . So that, if the diameter a b of the 
circle = a, the absciss a p = x, the ordinate PL = y ; then is 
x : a — x : : y z : x*, or x*=(a — x) y a , which is the equation to 
the curve. 

5. The right line e b e is an asymptote to the curve. 

6. Arch a m of the circle=arch b r, and arch a m=B r. 

7. The whole infinitely long cissoidal space, contained be- 
tween the asymptote e b e and the curves n o l a, &c. alow, 
&c. is equal to three times the area of the generating circle 

AOBO. 



Section III. — The Cycloid. 

The cycloid, or trochoid, is an elegant mechanical curve first 
noticed by Descartes, and an account of it was published by 
Mersenne in 1615. It is, in fact, the curve described by a nail 
in the rim of a carriage-wheel while it makes one revolution on 
a flat horizontal plane. 




CURVES : CYCLOID. 



171 




1. Thus, if a circle epp, keeping always in the same plane, 
be made to roll along the right line a b, until a fixed point p, in 
its circumference, which at first touched the line at a, touches it 
again after a complete revolution at b ; the curve apvpb de- 
scribed by the motion of the point p is called a cycloid. 

2. The circle e p f is called the generating circle ; and the 
right line a b, on which it revolves, is called the base of the 
cycloid. 

Also, the right line, or diameter, c v, of the circle, which bi- 
sects the base a b at right angles, is called the axis of the cycloid ; 
and the point v where it meets the curve, is the vertex of the 
cycloid. 

3. If p be a point in the fixed 
diameter a f produced, and the 
circle A e f be made to roll 
along the line a b as before, so 
that the point a, which first 
touches it at one extremity, 
shall touch it again at b, the 
curve p v p, described by the 
point p, is called the curtate 
cycloid. 

4. And, if the point p 
be any where in the un- 
produced diameter ap, 
and the circle a e f be 
made to roll along ab 
from a to b ; the curve 
p v p is, in that case, 

called the inflected or prolate cycloid. 

The following are the chief properties of the common cy- 
cloid. 

1. The circular arc v e = the 
line e g between the circle and 
cycloid, parallel to a b. 

2. The semicircumf. v e c=the 
semibase c b. 

3. The arc v g=2, the corres- 
ponding chord v e. 

4. The semicycloidal arc v G b=2 diam. vc. 

5. The tangent t g is parallel to the chord v e. 

6. The radius of curvature at v=2 c v. 

7. The area of the cycloid a v b c a is triple the circle cev; 
and consequently that circle and the spaces vec bg, ve'cag', 
are equal to one another. 





172 



CURVES : QUADRATRIX. 




8. A body falls through any arc l k of a cycloid reversed, 
in the same time, whether that 
arc be great or small ; that is, 
from any point l, to the lowest 
point k, which is the vertex re- 
versed : and that time is to the 
time of falling perpendicularly 
through the axis m k, as the se- 
micircumference of a circle is to 
its diameter, or as 3*141593 to 
2. And hence it follows that if 

a pendulum be made to vibrate in the arc l k n of a cycloid, all 
the vibrations will be performed in the same time. 

9. The evolute of a cycloid is another equal cycloid, so that if 
two equal semicycloids op,oq, be joined at o, so that o m be 
=m k the diameter of the generating circle, and the string of a 
pendulum hung up at o, having its length = o k or = the curve 
o p ; then, by plying the string round the curve o p, to which it 
is equal, if the ball be let go, it will describe, and vibrate in the 
other cycloid pkq; where o p=q k, and o q=p k. 

10. The cycloid is the curve of swiftest descent : or a heavy 
body will fall from one given point to another, by the way of 
the arc of a cycloid passing through those two points, in a less 
time, than by any other route. Hence this curve is at once in- 
teresting to men of science and to practical mechanics. 



Section IV. — The Quadratrix. 

The quadratrix is a species of curve by means of which the 
quadrature of the circle and other curves is determined mechani- 
cally. For the quadrature of the circle, curves of this class 
were invented by Dinostrates and Tschirnhausen, and for that 
of the hyperbola by Mr. Perks. We shall simply describe in 
in this place the quadratrix of Tschirnhausen ; and that in or- 
der to show its use in the division of an arc or angle. 



CURVES : QUADRATRIX. 



173 




To construct this quadratrix, divide the quadrantal arc A b 
into any number of equal parts, a n, n n, 
n n', n b ; and the radius a c into the same 
number of equal parts a p, p p, p p', p' c. 
Draw radii c n, c n, &c. to the points of di- 
vision upon the arc ; and let lines p m, p m, 
&c. drawn perpendicularly to a c from the 
several points of division upon it, meet the 
radii in m, m, m', &c. respectively. The 
curve a m m m' d that passes through the points of intersection 
M, ra, &c. is the quadratrix of Tschirnhausen. 

The figure a c d m' m m a thus constructed may be cut out from 
a thin plate of brass, horn, or pasteboard, and employed in the 
division of a circular arc. 

Thus, suppose the arc i l or the angle ikiis to be divided 
into five equal parts. Apply the side a b of the quadratrix upon 
i k, the point b corresponding with the 
angle k. Draw a line along the curve a s, 
cutting k l in p. Remove the instrument, 
and from p let fall the perpendicular p e 
upon i k. Divide e i into five equal parts 
by prob. 5, Practical Geometry, and 
through the points of division draw c m, 
d n, &c. parallel to e p. Then from their 
intersections, m, n, o, p, draw the lines 
k m, k n, k o, k p, and they will divide 
the angle ikl into five equal parts, as required. 

Note 1. — If, instead of dividing the arc into equal parts, it 
were proposed to divide it into a certain number of parts having 
given ratios to each other ; it would only be necessary to di- 
vide e i into parts having the given ratio, and proceed in other 
respects as above. 

Note 2. — If the arc or angle to be divided exceed 90 degrees, 
bisect it, divide that bisected arc or angle into the proposed 
number of parts, and take two of them for one of the required 
divisions of the whole arc. 




24 



Q2 



174 



CURVES : CATENARY. 



Section V. — The Catenary, and its application. 



The catenary is a mechanical curve, being that which is as- 
sumed by a chain or cord of uniform substance and texture, 
when it is hung upon two points or pins of suspension (whether 
those points be in a horizontal plane or not), and left to adjust 
itself in equilibrio in a vertical plane. 

This curve is of great interest to practical men on account of 
its connexion with bridges of suspension, or chain bridges. Its 
consideration cannot, therefore, with propriety be omitted, 
although it involves mechanical propositions which will be 
announced subsequently. 

Let a, b, be the points of suspension of such a cord, a a c b b 
the cord itself when hanging at rest in a vertical position. Then 
the two equal and symmetrical portions a a c, c b b, both ex- 
posed to the force of gravity upon every 
particle, balance each other precisely at c. 
And, if one half, as c b b, were taken away, 
the other half, a a c, would immediately 
adjust itself in the vertical position under 
the point a were it not prevented. Sup- 
pose it to be prevented by a force acting 
horizontally at c, and equal to the weight 
of a portion of the cord or chain equal in 
length to c m; then is c m the measure of the tension at the ver- 
tex of the curve ; it is also regarded as the parameter of the 
catenary. Whether the portion a a c hang from a, or a shorter 
portion, as a c, hang from «, the tension at c is evidently the 
same : for in the latter case the resistance of the pin at a, ac- 
complishes the same as the tension of the line at a when the 
whole a a c hangs from a.* 

Let the line c m which measures the tension at the vertex be 
=jo, let c d=x, a d—d b=y, c a — c b=z, c d = h, a b = d, 
c a A=c b B=| /. Then 

* This may easily be determined experimentally, by letting the cord hang very 
freely over a pulley at c, and lengthening or shortening the portion there suspend- 
ed, until it keeps a a c in its due position ; then is the portion so hanging beyond 
the pulley equal in length to c m. 




curves: catenary. 175 



. , p 4- x-\- \/2 p x + x 2 , 

1. y =^xhyp. log. -£ — £ v —=p x hyp. 

log. ■ " "*" g + z = jo x hyp. log. £ = jo m log. tan 

(45° + \ s). 

2. If the angle s of suspension made between the tangent 
to the curve at a or b, and the horizon be 45° ; then d : I : : 
1 : 1-1346. 

3. When 1=2 d, then h = '7966 d, and s = 77° 3'. 

4. When the angle s of suspension is 56° 28', then p, x, y, 
and z, are as 1, 0*81, 1-1995, and 1-5089 respectively. In this 
case t, the tension, at the point of suspension, is a minimum 
with respect to y. 

d 

5. Generally -j- = — tan s m l tan § s. 

Where m = 2-30255851, Napier's logarithm of 10. 

Or, log. j = log. tan s + log. (log. cot i s — 10) + -3622157 

— 10. This last formula serving to compute an approximative 
result. 

6. The distance of the centre of gravity of the whole curve 

2 z, from the vertex = £ (x + *—*- — p>) 

(p-x-x) 2 t z 

7. The radius of curvature — '— = — : this at the vertex 

P P 

is rad curv.=p. 

8. When s and p are given ; then 

z = p tan s . . . . t = p sec s. 

, , . p versin s 

x = p (sec s — 1 ) = -£ 

* ' cos s 

and y = p m log. tan (45° -f i s). 

9. When s and z, or § I are given : then 
p = z cots .... t = z cosec s 

x == z cosec s versin s 

y = m z cot s log. tan (45° + \ s). 

10. When s and y are given : then 
p=y -7-m log. tan (45° + § s) 
t=y-^u cos s log. tan (45° + h s) 
z=y tan s-~m log. tan (45° + 3 s) 

x=y versin s ■+■ m cos s log. tan (45° + 5 s). 



176 



CURVES : CATENARY. 



11. When x and y are given ; then 

log. tan (45° + i s) _ y 
sec s versin s m x ' 

from which s may be found by an approximative process ; also 
x x x sin s 

jD = ..../ = ; Z= : 

sec s — 1 versin s versin s 

in all these cases t is determined in length of chain or cord of 
which the catenary is actually constructed. 

12. To draw the catenary mechanically. — If the dis- 
tance a b between the points of suspension, and the depth d c 
of the lowermost point, be given (see the preceding figure), 
hang one extremity of a fine uniform chain or cord at one of 
the points a, and, letting the chain or cord adjust itself as 
a festoon in a vertical plane, lengthen or shorten it as it is 
held near the other end, over a pin at b, until, when at rest, 
it just reaches the point c ; so shall the cord form the cate- 
nary ; and a pencil passing along the cord, from a by a, c, b, to 
b, will mark the curve upon a vertical board brought into con- 
tact with it. 

13. All catenaries that make equal angles with their ordi- 
nates at their points of suspension are similar, and have x to y 
a constant ratio : and of any two which do not make equal 
angles, but have x to y in different ratios, 

a portion may be cut from one curve 
similar to the other. Thus, let a c b and 
a' c' b', be the two curves, of which a' c' b' 
is the flattest. Suppose them placed upon 
one axis d c c', and the tangent t' s', to 
the lower curve at b', the point of suspen- 
sion to be drawn. Then, parallel to t' s' 
draw another line t s to touch the other 
curve in b. Through b draw b a parallel 
to b' a'. So shall the portion a c b of the 
upper catenary be similar to the lower cate- 
nary a' c' b'. 

14. With reference to the practical A 
uses of the catenary, we may now blend 
the geometrical and the mechanical 
consideration of its properties. Taking 
any portion c b of the catenary, from 
the lowest point c ; its weight may be 
regarded as supported by tensions act- 
ing in the tangential directions c n, b n. 
The strains at c and b may be conceived 
as acting at the point of intersection n ; 




\ / 


/ 


V 




,r 


~""\/n 


T ^ 


ft \ 

J 



CURVES : CATENARY. 177 

above which, therefore, in the vertical direction n g, the weight 
of the portion c b may be conceived to act at its centre of gra- 
vity G. 

Hence, strain at c : weight of c b :: sin g n b : sin b n c 

:: cos b n r : sin b n r 
: : radius : tan b n r 
:: radius : tan d b n 
Hence, the horizontal tension at c being constant, the weight, 
and consequently the length of any portion c b of the uniform 
chain must be proportional to the tangent of the inclination of 
the catenary to the horizon at the extremity b of the said portion. 
This may be regarded as the characteristic property of the 
catenary. 

15. In like manner, 

hor. strain at c : oblique strain at b : : sin b n g : sin c n g 

:: cos b n r : radius 
: : rad : sec Jnr. 
Therefore, the strain exerted tangentially, at any point b, is 
proportional to the secant of the inclination at that point. 
Also, from (14) and (15) 

tang, strain at b : weight of b b c : : sec d b n : tan d b n. 
These properties evidently accord with the preceding equa- 
tions. 

16. Let, then, c o, in the axis produced downwards, be equal 
to the parameter or the measure of the horizontal strain at c ; 
and upon o as a centre with radius c o describe a circle. A tan- 
gent d t drawn to this circle from d, will be parallel to the tan- 
gent b n h of the curve at the point b to which d b is the ordi- 
nate. That tangent d t (to the circle) will also be equal in 
length to the corresponding portion b c of the curve : while 
the tension at b will be expressed by a length of the chain 
equal to the secant o d. So again, if d t be a tangent, to the 
circle drawn from d, it will be equal in length to b b c, and 
parallel to the tangent to the catenary at b ; while the secant 
o d will measure the oblique tension at b ; evidently exceed- 
ing the constant horizontal tension or strain at c, by the ab- 
sciss c D. 

17. When the parameter of the catenary, or the line which 
measures the tension at the lowest point, is equal to the depres- 
sion d c ; if each of these be supposed equal to 1, then a b = 
2-6339, the length of chain acb = 3-4641 ; the strain at the 
points of suspension a and b will each be 2, that at the lowest 
point being 1 ; and the chain at a and b will make an angle of 
60° with the horizon. 

18. If the strain at c be equal to the weight of the chain, 
and each denoted by 1: then a b = -96242, d c = -1180340^ 



178 curves: catenary. 

the tension at a or b = 1*118, the angle of suspension at those 
points nearly 26° 34' ; the width of the curve is 8*1536 times, 
and the length 8*4719 times, the depression d c. 

19. If the strain or tension at the lowest point be double the 
weight of the chain : then if the parameter be 1, a b will be 
•49493, c D = -03078, the strain at a or b 1*03078, the angle 
of suspension about 14° 2', the width or span 16*0816 times, 
and the length of chain 16*2462 times the depression. 

The magnitudes of the lines, angles, and strains in many other 
cases may be seen in the table below. The whole theory may 
be verified experimentally, by means of spring steel-yards ap- 
plied to a chain of given length and weight, placed in various 
positions ; according to the method suggested subsequently 
when treating of the parallelogram of forces, in Mechanics. 

20. Taking a b = d, c d = A, length a c b = /, strain at c or 
parameter = p, then, in all cases where the depression is small 
compared with the length of the chain, Prof essor Leslie shows,* 
that 

d* . , r • d* , _ , 

p = — —4- 1 h .... strain at a or b =*= — — + f- h 
8/i 8 /i 

/a la 

orp = — § h . . . . strain at a or b = ■— — + \ h 

8 h 8 h, 

In this case, too, the strains at c and a or b are nearly in the 
inverse ratio of the depression.! 

* Elements of Natural Philosophy, p. 63. 

f For a very complete investigation of the proper forms of catenaries for suspen- 
sion bridges, with remarks on the Menai bridge, and on the failure of the suspen- 
sion bridge at Broughton, see Mr. Eaton Hodgkinson's paper in the Memoirs of 
the Manchester Society, vol. v., New Series. 



curves: catenary. 



179 



Table of Relations of Catenarian Curves, the Parameter 
being denoted by 1. 



Angle of 
suspen- 
sion. 


DC 


DB 


C B 


tension at 
AorB 


DB-t-DC 


1° 0' 


•00015 


•01745 


•01745 


1-0001 


114-586 


2 


•00061 


•03491 


•03492 


1-0006 


57-279 


3 


•00137 


•05238 


•05241 


1-0014 


38-171 


4 


•00244 


•06987 


•06993 


1-0024 


28-613 


5 


•00382 


•08738 


•08749 


1-0038 


22-874 


6 


•00551 


•10491 


•10510 


1-0055 


19-046 


7 


•00751 


•12248 


•12278 


1-0075 


16-309 


8 


•00983 


•14008 


•14054 


1-0098 


14-254 


9 


•01247 


•15773 


•15838 


1-0125 


12-654 


10 


•01543 


•17542 


•17633 


1-0154 


11-372 


11 


•01872 


•19318 


•19438 


1-0187 


10-820 


12 


•02234 


•21099 


•21256 


1-0223 


9-444 


13 


•02630 


•22887 


•23087 


1-0263 


8-701 


14 


•03061 


•24681 


•24933 


1-0306 


8-062 


15 


•03528 


•26484 


•26795 


1-0353 


7-508 


16 


•04030 


•28296 


•28675 


1-0403 


7-021 


17 


•04569 


•30116 


•30573 


1-0457 


6-591 


18 


•05146 


•31946 


•32492 


1-0515 


6-208 


19 


•05762 


•33786 


•34433 


1-0576 


5-863 


20 


•06418 


•35637 


•36397 


1-0642 


5-553 


21 


•07114 


•37502 


•38386 


1-0711 


5-271 


22 


•07853 


•39376 


•40403 


1-0786 


5-014 


23 


•08636 


•41267 


•42447 


1-0864 


4-778 


24 


•09484 


•43169 


•44523 


1-0946 


4-562 


25 


•10338 


•45087 


•46631 


1-1034 


4-361 


26 


•11260 


•47021 


•48773 


1-1126 


4-176 


28 


•13257 


•50940 


•53171 


1-1326 


3-843 


30 


•15470 


•54930 


•57735 


1-1547 


3-551 


32 4 


•18004 


•5912 


•62649 


1-1800 


3-284 


34 16 


•21003 


•6371 


•68130 


1-2100 


3-034 


36 52 


•24995 


•6932 


•74991 


1-2499 


2-773 


39 11 


•29011 


•7443 


•81510 


1-2901 


2-567 


41 44 


•34004 


•8029 


•89201 


1-3400 


2-362 


44 


•39016 


•8566 


•96569 


1-3902 


2-196 


46 1 


•43999 


•9066 


1-0361 


1-4400 


2-060 


48 11 


•49981 


•9623 


1-1178 


1-4998 


1-925 


50 8 


•56005 


1-0142 


1-1974 


1-5800 


1-811 


52 9 


•62973 


1-0706 


1-2869 


1-6297 


1-699 


54 13 


•71021 


1-1304 


1-3874 


1-7102 


1-592 


56 28 


•81021 


1-1995 


1-5089 


1-8102 


1-481 


58 3 


•88972 


1-2510 


1-6034 


1-8897 


1-416 


60 


1-0000 


1-3169 


1-7321 


2-0000 


1-317 


64 6 


1-2894 


1-4702 


2-0594 


2-2894 


1-140 


67 28 


1-6095 


1-6135 


2-4102 


2-6095 


1-002 


67 32 


1-6168 


1-6164 


2-4182 


2-6168 


0-9998 



180 CURVES I CATENARY. 

21. The preceding table is abridged from a very extensive 
one given by Mr. Ware in his " Tracts on Vaults on Bridges." 
Two examples will serve to illustrate its use. 

Ex. 1. Suppose that the span of a proposed suspension bridge 
is 560 feet, and the depression in the middle 251 feet ; what will 
be the length of the chain, the angle of suspension at the extre- 
mities, and the ratio of the horizontal pressure at the lowest 
point, and the oblique pressures at the points of suspension, with 
the entire weight of the chain ? 

Here d b-*-d c=280-j-25*875=10*82, a number which is to 
be found in the table. 

Opposite to that number, we find 11° for the angle of suspen- 
sion, d b=*19318, c b=*19438, tension at a or b=1-0187, the 
constant tension at the vertex being 1. (fig. p. 174.) 

Consequently, -19318 : -19438 :: 560 : 563-48 length of the 
chain. 

Also, horizon, pressure at c are as 1*0000 

oblique pressure at a or b ... . 1-0187 
entire weight of chain .... and -39876 

Ex. 2. Suppose that while the span remains 560, the depres- 
sion is increased to 51. 

HereD b-*-d c=280-*-51=5*49. This number is not to be 
found exactly in the table. The nearest is 5*553 in the last 
column, agreeing with 20°, the angle of suspension. 

Now, 5-55 — 5*49 = -06, and 5-55 — 5*27 = -28, the former 
difference being nearly one-fifth of the latter. Hence, adding 
to each number, in the line agreeing with 20°, one- fifth of the 
difference between that and the corresponding number in the 
next line, we shall have 

Angle of suspension = 20° 12', d c = '06556, db = -36010, 
c d=-36797, tension at a=1'10656. 

Hence -36001 : -36797 :: 560 : 572-24, length of chain. 

Also, horizontal pressure at c are as 1 -0000 

oblique pressure at a or b — 1*10656 

entire weight of chain -73594 

Comparing this with the former case, it will be seen that the 
tensions at c and a, in reference to the weight of the chain, are 
diminished nearly in the inverse ratio of the two values of d c ; 
thus confirming the remark in art. 20. 

In practical cases with regard to bridges of suspension, it will 
be easy, when the weight of the material and its cohesive 
strength are known, to find the relative strength of any pro- 
posed structure. 



ISOMETRICAL PERSPECTIVE. 181 



CHAPTER VII. 
Professor Parish's Isometrical Perspective. 

In the course of lectures which I deliver in the university 
of Cambridge, I exhibit models of almost all the more im- 
portant machines which are in use in the manufactures of 
Britain. 

The number of these is so large, that had each of them been 
permanent and separate, on a scale requisite to make them work, 
and to explain them to my audience, I should, independently 
of other objections, have found it difficult to have procured a 
warehouse large enough to contain them. I procured therefore 
an apparatus, consisting of what may be called a system of the 
first principles of machinery ; that is, the separate parts, of 
which machines consist. These are made chiefly of metal, so 
strong, that they may be sufficient to perform even heavy work : 
and so adapted to each other, that they may be put together at 
pleasure, in every form, which the particular occasion re- 
quires. 

Those parts are various : such as, loose brass wheels, the 
teeth of which all fit into one another : axes, of various lengths, 
on any part of which the wheel required may be fixed : bars, 
clamps, and frames ; and whatever else might be necessary 
to build up the particular machines which are wanted for 
one lecture. These models may be taken down, and the 
parts built up again, in a different form, for the lecture of 
the following day. As these machines, thus constructed for 
a temporary purpose, have no permanent existence in them- 
selves, it became necessary to make an accurate representa- 
tion of them on paper, by which my assistants might know 
how to put them together without the necessity of my con- 
tinual superintendance. This might have been done, by 
giving three orthographic plans of each ; one on the horizon- 
tal plane, and two on vertical planes at right angles to each 
other. But such a method, though in some degree in use 
among artists, would be liable to great objections. It would 
25 R 



182 ISOMETRICAL PERSPECTIVE. 

be unintelligible to an inexperienced eye ; and even to an artist, 
it shows but very imperfectly that which is most essential, the 
connexion of the different parts of the engine with one another ; 
though it has the advantage of exhibiting the lines parallel to 
the planes on which the orthographic projections are taken 
on a perfect scale. 

This will be easily understood, by supposing a cube to be 
the object represented. The ground plan would be a square 
representing both the upper and lower surfaces. And the two 
elevations would also be squares on two vertical planes, parallel 
to the other sides of the cube. The artist would have exhibited 
to him three squares ; and he would have to discover how to 
put them together in the form of a cube, from the circumstance 
of there being two elevations and a ground plan. This method, 
therefore, giving so little assistance on so essential a point, I 
thought unsatisfactory. 

The taking a picture on the principles of common perspec- 
tive, was the next expedient that suggested itself. And this 
might be adapted to the exhibition of a model, by taking a 
kind of bird's-eye view of the object, and having the plane of 
the picture, not as is most common in a drawing, perpendicular 
to the horizon, but to a line drawn from the eye, to some prin- 
cipal part of the object. For example : in taking the picture 
of a cube, the eye might be placed in a distant point on the line 
which is formed by producing the diagonal of the cube. But 
to this common perspective there are great objections. The 
lines, which in the cube itself are all equal, in the representation 
are unequal. So that it exhibits nothing like a scale. And 
to compute the proportions of the original from the repre- 
sentation would be exceedingly difficult, and, for any useful 
purpose, impracticable : there is equal difficulty too, in com- 
puting the angles which represent the right angles of the cube. 
Neither does the representation appear correct, unless the 
eye of the person, who looks at it, be placed exactly in the 
point of sight. It is true that, as we are continually in the 
habit of looking at such perspective drawings, we get the 
habit of correcting, or rather overlooking the apparent errors 
which arise from the eye being out of the point of sight, 
and are therefore not struck with the appearance of incorrect- 
ness, which if we were unaccustomed to it, we should feel 
at once. 

The kind of perspective which is the subject of this pa- 
per, though liable in a slight degree to the last mentioned in- 
convenience, till the eye becomes used to it, I found much bet- 
ter adapted to the exhibition of machinery ; I therefore deter- 



ISOMETRIC AL PERSPECTIVE. 183 

mined to adopt it, and set myself to investigate its principles, 
and to consider how it might most easily be brought into 
practice. 

It is preferable to the common perspective on many accounts, 
for such purposes. It is much easier and simpler in its princi- 
ples. It is also, by the help of a common drawing-table, and 
two rulers,* incomparably more easy, and, consequently, more 
accurate in its application ; insomuch, that there is no difficulty 
in giving an almost perfectly correct representation of any 
object adapted to this perspective, to which the artist has access, 
if he has a very simple knowledge of its principles, and a little 
practice. 

It further represents the straight lines which lie in the 
three principal directions, all on the same scale. The right 
angles contained by such lines are always represented either 
by angles of 60 degrees, or the supplement of 60 degrees. 
And this, though it might look like an objection, will appear 
to be none on the first sight of a drawing on these principles, 
by any person who has ever looked at a picture. For, he 

* It is unnecessary to describe the drawing-table any further than by observing 
that it ought to be so contrived, as to keep the paper steady on which the drawing 
is to be made. 

Here should be a ruler in the form of the letter T to slide on one side of the 
drawing-table. The ruler should be kept, by small prominences on the under side, 
from being in immediate contact with the paper, to prevent its blotting the fresh 
drawn lines as it slides over them. And a second ruler, by means of a groove near 
one end on its under side, should be made to slide on the first. The groove should 
be wider than the breadth of the first ruler, and so fitted, that the second may at 
pleasure be put into either of the two positions represented in the plate, fig. 1, so as 
to contain with the former ruler, in either position an angle of 60 degrees. The 
groove should be of such a size, that when its shoulders a and d are in contact with, 
and rest against the edges of the first ruler, the edge of the second ruler should 
coincide with de, the side of an equilateral triangle described on d g; a. portion of 
the edge of the first ruler ; and when the shoulders b and c rest against the edges 
of the first ruler, the edge of the second should lie along g e, the other side of the 
equilateral triangle. The second ruler should have a little foot at k for the same 
purpose as the prominences on the first ruler, and both of them should have their 
edges divided into inches, and tenths, or eighths of inches. 

It would be convenient if the second ruler had also another groove r s, so formed 
that when the shoulders r and s are in contact with the edges of the first ruler, the 
second should be at right angles to it. For representing circles in their proper 
positions, the writer made use of the inner edge of rims cut out from cards, into 
isometrical ellipses as represented in the figure ; of these he had a series of different 
sizes, corresponding to his wheels. Such a series might be cut by help of the con- 
centric ellipses in fig. 5, but he thinks that it would be an easier way to make use 
of that set of concentric ellipses as they stand, by putting them in the proper place 
under the picture, if the paper on which the drawing is made be thin enough for 
the lines to be traced through, as by the help of them the several concentric circles 
will go to the representation of one which might be drawn at once. It is difficult to 
execute them separately with sufficient accuracy to make them correspond. For 
this purpose a separate plate of fig. 5 should be had, and one edge of the paper on 
the drawing-table should be loose to admit of the concentric ellipses being slid under 
it to the proper place, as described p. 187. 



184 ISOMETRIC AL PERSPECTIVE. 

cannot for a moment have a doubt, that the angle represented is 
a right angle, on inspection. 

And we may observe further, that an angle of 60 degrees is 
the easiest to draw, of any angle in nature. It may be instantly 
found by any person who has a pair of compasses, and under- 
stands the first proposition of Euclid. The representation, 
also, of circles and wheels, and of the manner in which they act 
on one another is very simple and intelligible. The principles 
of this perspective, which, from the peculiar circumstance of its 
exhibiting the lines in the three principal dimensions on the 
same scale, I denominate " Isometrical" will be understood 
from the following detail : 

Suppose a cube to be the object to be represented. The eye 
placed in the diagonal of the cube produced. The paper, on 
which the drawing is to be made to be perpendicular to that 
diagonal, between the eye and the object, at a due proportional 
distance from each, according to the scale required. Let the 
distance of the eye, and consequently that of the paper, be in- 
definitely increased, so that the size of the object may be in- 
considerable in respect of it. 

It is manifest, that all the lines drawn from any points of 
the object to the eye may be considered as perpendicular to 
the picture, which becomes, therefore, a species of orthogra- 
phic projection. It is manifest, the projection will have for 
its outline an equiangular and equilateral hexagon, with two 
vertical sides, and an angle at the top and bottom. The other 
three lines will be radii drawn from the centre to the lowest 
angle, and to the two alternate angles ; and all these lines 
and sides will be equal to each other both in the object and 
representation : and if any other lines parallel to any of 
the three radii should exist in the object, and be represented 
in the picture, their representations will bear to one another, 
and to the rest of the sides of the cube, the same propor- 
tion which the lines represented bear to one another in the 
object. 

If any one of them, therefore, be so taken as to bear any 
required proportion to its object, e. g. 1 to 8, as in my repre- 
sentations of my models, the others also will bear the same pro- 
portion to their objects ; that is, the lines parallel to the three 
radii will be reduced to a scale. 

I omit the demonstration of this, and some other points, 
partly for the sake of brevity, and partly because a geometri- 
cian will find no diificulty in demonstrating them himself 
from the nature of orthographic projection ; and a person, 
who is not a geometrician, would have no interest in reading a 
lemonstration. 



ISOMETRICAL PERSPECTIVE. 1S5 

For the same reason, it is unnecessary to show that the 
three angles at the centre are equal to one another, and each 
equal to 120 degrees, twice the angle of an equilateral tri- 
angle ; and the angle contained between any radius and side 
is 60 degrees, the supplement of the above, and equal to the 
angle of an equilateral triangle. All this follows immediately 
from Euclid, B. IV. Prop. 15, on the inscription of a hexagon 
in a circle. 

In models and machines, most of the lines are actually in the 
three directions parallel to the sides of a cube, properly placed 
on the object. And the eye of the artist should be supposed to 
be placed at an indefinite distance, as before explained, in a 
diagonal of the cube produced. 



Definitions . 

The last mentioned line may be called the line of sight. 

Let a certain point be assumed in the object, as for example 
c, fig. 2, pi. I. and be represented in the picture, to be called 
the regulating point. Through that point on the picture may 
be drawn a vertical line c e, fig. 2, and two others, c b, c g, 
containing with it, and with one another, angles of 120°, to 
be called the isormtrical lines, to be distinguished from one 
another by the names of the vertical, the dexter, and the 
sinister lines. And the two latter may be called by a common 
name — the horizontal isometrical lines. Any other lines, 
parallel to them, may be called respectively by the same 
names. The plane passing through the dexter and vertical 
lines may be called the dexter isometrical plane ; that passing 
through the vertical and sinister lines, the sinister plane ; and 
that through the dexter and sinister lines, the horizontal 
plane. 

By the use of the simple apparatus described above in the 
note, the representation of these lines in the objects may be 
drawn on the picture, and measured to a scale, with the 
utmost facility, the point at the extremity being first found 
or assumed. The position of any point in the picture may 
be easily found, by measuring its three distances, namely, 
first its perpendicular distance from the regulating horizontal 
plane (that is, the horizontal plane passing through the regu- 
lating point), secondly, the perpendicular distance of that 
point where the perpendicular meets the horizontal plane, 
from the regulating dexter line ; and thirdly, of the point, 
where that perpendicular meets the dexter line from the 
regulating point ; and then taking those distances reduced to 

R 2 



186 ISOMETRICAL PERSPECTIVE. 

the scale, first, along the dexter line ; secondly, along the sin 
ister line ; and thirdly, along the vertical line, in the picture. 
These three may be called the dexter distance of the point, its 
sinister distance, and its altitude. And it is manifest they 
need not be taken in this order, but in any other that may be 
more convenient to the artist, there being six ways in which 
this operation may be varied. 

If any point in the same isometrical plane, with the point re- 
quired to be found, is already represented in the picture, that 
point may be assumed as a new regulating point, and the point 
required found by taking two distances ; and if the new assumed 
regulating point is in the same isometrical line with the point, 
it is found by taking only one distance. And this last simple 
operation will be found in practice all that is necessary for the 
determination of most of the points required. Thus any paral- 
lelopiped, or any frame work, or other object with rafters, or 
lines lying in the isometrical directions, may be most easily and 
accurately exhibited on any scale required. But, if it be neces- 
sary to represent lines in other directions, they will not be on 
the same scale, but may be exhibited, if straight lines, by find- 
ing the extremities as above, and drawing the line from one to 
the other ; or sometimes more readily in practice by help of an 
ellipse, as hereafter described. 

If a curved line be required, several points may be found 
sufficient to guide the artist to that degree of exactness which is 
required. 

The method of exhibiting the representations of any ma- 
chines, or objects, the lines of which lie, as they generally do, in 
the isometrical directions ; that is, parallel to the three direc- 
tions of the lines of the cube, is as has been already shown ; and 
likewise the mode of representing any other straight lines, by 
finding their extremities ; or curved lines, by finding a number 
of points. 

But in representing machines and models, there are not only 
isometrical lines, but also many wheels working into each other, 
to be represented. These, for the most part, lie in the 
isometrical planes ; and it is fortunate that the picture of a cir- 
cle in any one of these planes is always an ellipse of the 
same form, whether the plane be horizontal, dexter, or sin- 
ister ; yet they are easily distinguished from each other by the 
position in which they are placed on their axle, which is an 
isometrical line, always coinciding with the minor axis of the 
ellipse. 

This will be obvious from considering the picture of a cube 
with a circle inscribed in each of its planes, fig. 3, and 
considering these circles as wheels on an axle. The two other 



ISOMETRICAL PERSPECTIVE. 187 

lines (or spokes of the wheel) in the ellipse, which are drawn 
respectively through the opposite points of contact of the circle 
with the circumscribing figure, are isometrical lines also ; for 
the points of contact bisect the sides of the circumscribing 
parallelogram, and therefore the lines are parallel to the 
other sides. They give likewise the true diameter of the 
wheels, reduced to the scale required. It further appears, 
from the nature of orthographic projection, that the major 
axis of the ellipse is to the minor axis as the longer to the 
shorter diagonal of the circumscribing parallelogram, that is 
(since the shorter diagonal divides it into two equilateral 
triangles) as the square root of three to one ; as appears from 
Euclid, Lib. I. Prop. 47 ; and since the sum of the squares 
of the conjugate diameters in an ellipse, is always the same, 
if we put v^l for the minor axis, the v/3 for the major, and i 
for the isometrical diameter, we shall have 2 i 2 = 1 -f 3, = 4, 
and i = \/2. 

Therefore the minor axis, the isometrical diameter, and the 
major axis, may be represented respectively by ^/l, \/2, \/3, 
or nearly by 1, 1-4142, 1*7321 ; or more simply, though not so 
nearly, by 28, 40, 49. 

These lines may be geometrically exhibited by the follow- 
ing construction: 

Let a b, fig. 4, be equal to b d, and the angle at b, a right 
angle. In b a produced, take b a = to ad, draw a d, and 
produce both it, and a b. Then will b d, b a, and a d, be 
respectively to one another, as x/1, </2, ^/S, by Euclid, I. 47. 
Therefore if a /3 be taken equal to the isometrical diameter of 
the ellipse required, j3 5 drawn perpendicular to it will be the 
minor axis, and a 8 the major axis. The ellipse itself, there- 
fore, may be drawn by an elliptic compass, as that instru- 
ment may be properly set, if the major and minor axis are 
known. If it is to represent a wheel on an axle, care must be 
taken to make the minor axis lie along that axle. In the ab- 
sence of the instrument it may be drawn from the concentric 
ellipsis, fig. 5, which may be placed under the paper, in the 
position above described, and seen through it, if the paper 
be not too thick ; and in this method the smaller concentric 
circles of the wheel may be described at the same time, as they 
may be seen through the paper, or if they should not be ex- 
actly of the right size, it would be easy to describe them by 
hand between the two nearest concentric ellipses ; and thus 
also the height of the cogs of a wheel in the different parts 
of it may be exhibited longer and narrower towards the ex- 
tremities of the minor axis. Their width may be determined 
from the divisions of the ellipse. In most cases this may be 



188 ISOMETRIC AL PERSPECTIVE. 

done with sufficient accuracy from the circumference of the el- 
lipse being divided into eight equal divisions of the circle, by 
the two axes, and two isometrical diameters, each of which parts 
may be subdivided by the skill of the artist ; and not only the 
face of the wheel in front may be thus exhibited, but the parts of 
the back circles also, which are in sight, may be exhibited by 
pushing back the system of concentric ellipses on the minor 
axis or axle through a distance representing the breadth of the 
wheel, and then tracing both the exterior and the interior circles 
of the wheel, and of the bush on which it is fixed, as far as 
they are visible. Care should be taken to represent the top 
of the teeth, or cogs, by isometrical lines, parallel to the axle, 
in a face wheel, or tending to a proper point in the axle in a 
bevil-wheel. And nearly in the same way may the floats of a 
water-wheel be correctly represented. If a series of concentric 
ellipses such as are given fig. 5, be not at hand, it will still be easy 
for an artist to draw the ellipses with sufficient accuracy for 
most purposes, by drawing through the proper point in the 
axle, the major and minor axes, and the two isometrical 
diameters, thus making eight points in the circumference to 
guide him. 

If in any case it should become necessary to represent 
a circle, which does not lie in an isometrical plane, we 
may observe that the major axis will be the same in what- 
ever plane it lies : and it will be the picture of that diameter, 
which is the intersection of the circle with the plane parallel to 
the picture, passing through its centre. And the major axis 
will bear to the minor axis the proportion of radius to the sine 
of the inclination of the line of sight to the plane of the circle. 
We may observe further, that the diameters of the ellipse, which 
are to the major axis as ^/2 to x/3, when such exist, are isome- 
trical lines.* 

And the representation of every other line, parallel and 
equal to any diameter of the circle, may be exhibited by draw- 
ing it equal and parallel to the corresponding diameter in the 

* We may remark, that if a cone be described, having its vertex at c 
which lies in the line of sight fig. 2, and passing through the three radii c b, c e> 
c g, all the straight lines in the superfices of that cone passing through c, and all 
other lines parallel to any of them, are isometrical, as well as those parallel to the 
three principal isometrical lines c b, c e, c g ; and no other lines but these can 
be on the same scale But though these multiply the number of isometrical 
lines infinitely, it is of little practical use, because it is only those which are parallel 
to the three principal lines, that can easily be distinguished at sight, to be isometrical. 

We may further remark, that if a line be drawn through the point c parallel to 
any given line whatever, and that line be made to revolve round the line of sight, 
at the same angular distance from it, so as to describe the surface of a cone, all 
other lines parallel to it, m any of its positions, will be isometrical, as they re- 
spect one another. 



ISOMETRICAL PERSPECTIVE. 189 

ellipse. If it should be desired to divide the circumference of 
an ellipse into degrees, or any number of parts representing 
given divisions of the circle, it may be done by the following 
method : 

Let an ellipse be drawn, fig. 6, and on its major axis, a g, a 
circle described, with its circumference divided into degrees or 
parts in any desired proportion, at b, c, d, e, f, &c, from which 
points draw perpendiculars to the major axis. They will cut 
the periphery of the ellipse in corresponding points. It would 
be difficult, however, in this way, to mark, with sufficient accu- 
racy, the degrees, which lie near the extremities of the major 
axis. But the defect may be supplied by transferring those 
degrees in a similar way, from a graduated circle, described on 
the minor axis. In this manner an isometrical ellipse may be 
formed into an isometrical circular instrument, or an isometrical 
compass, which may show bearings or measure angles on the pic- 
ture, in the same manner as a real compass or circular instru- 
ment would do in nature. 

It may be often useful to have a scale to measure distances, 
not only in the isometrical directions, but in others also. And 
this may be done by a series of similar concentric ellipses, as 
in fig. 7, dividing the isometrical diameters into equal portions. 
The other diameters will be so divided as to serve for a scale 
for all lines parallel to them respectively. 

Thus, in the isometrical squares, exhibited in fig. 2, distances 
measured on the longer diagonal, or its parallels, would be 
measured by the divisions on the major axis, those depending 
on the shorter diagonal by the divisions on the minor axis. 

To describe a cylinder lying in an isometrical direction, the 
circles at its extremities should be represented by the proper 
isometrical ellipses, and two lines touching both should be 
drawn : and in a similar way, a cone, or frustrum of a cone, 
may be described. A globe is represented by a circle whose 
radius is the semi-major axis of the ellipse representing a great 
circle. 

It would not be difficult to devise rules for the representation 
of many other forms which might occur in objects to be repre- 
sented. But the above cases are sufficient to include almost 
every thing which occurs in the representation of models, of 
machines, of philosophical instruments, and, indeed, of almost 
any regular production of art. 

Buildings may be exhibited by this perspective as correctly, 
in point of measurement, as by plans and elevations, under the 
advantage of having the full effect of a picture. 

A bridge, or any circular or gothic arch, consisting of por- 
26 



190 ISOMETRICAL PERSPECTIVE. 

tions of circles lying in isometrical planes, may be represented 
by portions of isometrical ellipses, which will easily be adapted 
and drawn upon the principles already explained, by which 
wheels are exhibited on their axles. The centres of those cir- 
cles must be found with which the centres of the ellipses must 
be made to coincide, their minor axes lying along the lines drawn 
from those centres perpendicular to the planes of the circles. 
The shaft of a pillar consists of a frustrum of a cone and a cylin- 
der united ; or perhaps of a cylinder alone, or a congeries of 
cylinders ; and we have already shown the method of exhibit- 
ing these, as well as their bases. And on the same principles, 
the position and size of the volutes and ornaments of the capi- 
tal may be found, and such guiding points as will make it easy 
to trace their forms. Thus the different courts and edifices 
of a cathedral, a college, or a palace, may be correctly depicted ; 
and even the rooms and internal structure, though less in the 
form of a picture, may be exhibited in such a way as to enable 
an architect, or his employer, to contemplate their situation, 
their ornaments, furniture, or any other circumstance belonging 
to their appearance, and to mark down exactly what he would 
have done, in such a way as could hardly be misunderstood by 
an attentive agent, though at a distance. 

But in thus exhibiting buildings as transparent, and their in- 
terior laid open, there is a danger of being confused by a multi- 
plicity of lines, which is a difficulty in a building containing 
many rooms that would need some address to get over. It is 
better adapted to exhibit the inside of a single room, of a ca- 
thedral, for instance, the aisles and transepts of which would not 
cause any great perplexity. 

In the same manner a plan of a city might be given, which 
would not only represent its streets and squares, as well (by the 
help of the scale above described, fig. 7,) as a common plan, but 
also a picture of its churches and public buildings, and even its 
private houses, if such were the design contemplated by the ar- 
tist, as they would almost all become visible when looked down 
upon from the commanding height which this perspective 
supposes. And such a single exhibition, if well executed, might 
give a better idea of a distant capital than a volume of descrip- 
tion. 

In the instances which have been given, most of the lines are 
isometrical. But the art is applicable to many cases, where 
there are few, or none such. It may be necessary, in many 
of them, to draw isometrical lines, or isometrical ellipses, 
by way of a guide, to determine the position of certain lines 
and points to enable the artist to describe with accuracy what 



ISOMETRICAL PERSPECTIVE. 191 

he has in view. And there is scarce any form so anomalous 
as to preclude the artist from taking advantage of these 
methods of ascertaining such lines or points in it as will 
give him much assistance in representing it with precision. 
If the intention be merely to make a picture, the guiding lines 
may be obliterated as soon as they have served the purpose de- 
signed, or they may be retained in some cases, and their 
lengths or diameters noted down in figures, if it be wished, 
to give ready information. And often, if the artist wishes 
to provide materials to enable him at his leisure to give accu- 
rate descriptions or exact drawings, the rudest exhibition of 
such lines may completely serve his purpose, provided he 
notes down on the spot such measurements with accuracy, 
however unexact the lines may be on which they are re- 
corded. In many cases it may be expedient to take liberties 
with this perspective, or with the picture, which will make it 
suit the purpose designed. And this will produce no confu- 
sion, provided those liberties are explained : for instance, it 
may often be expedient to make the scale in the vertical 
direction larger, sometimes very considerably so, than in the 
horizontal. It may in some cases be necessary to represent on 
paper what is hid in nature. What has been said on the 
internal structure of buildings is an instance of this, as well 
as what we shall observe on the exhibition of subterraneous ob- 
jects. We shall proceed to give some examples of these ob- 
servations. 

To give such a representation of an Etruscan vase, as 
would enable an artist to model it exactly, would be ex- 
ceedingly easy. Let a vertical line be drawn to represent 
the axis of the vase, fig. 8, and let points be taken in that axis, 
corresponding to the centres of the principal circles of the 
vase ; through which the horizontal isometrical lines may be 
drawn representing the radii of those circles, by the help of 
which the isometrical ellipses representing them are easily 
drawn. These will become a complete guide to the artist. He 
may assist himself by looking at the object along the line of 
sight, and then, if he has any skill in drawing, he will find no 
difficulty in tracing the outline from one of these to the other, 
with sufficient correctness. If he is unskilled in the art, of 
course he must be at the trouble of finding a larger number 
of ellipses to guide him. And in a similar manner, any solid 
formed by the revolution of a plane figure round one of its 
sides may be represented. 

The laying down the timbers of a ship, or making a picture 
of one, shall be another example. 



192 ISOMETRICAL PERSPECTIVE. 

Let a vertical isometrical plane be conceived to pass through 
its keel, and to be intersected by the perpendicular planes pass- 
ing through the ribs, and by planes parallel to the decks. The 
isometrical lines, which are the intersections of these, may be 
measured in the ship, and represented with their proper measures 
noted down in the picture, which will afford the means of re- 
presenting the ribs, and laying them down in their proper 
places. 

If this should be designed for the purpose of constructing a 
ship from a given model, it might be sufficient to represent 
the ribs only on one side ; those on the other side being the 
exact counterparts. If the purpose should be to make use of 
these lines for a drawing, they need be marked but very faintly, 
and the artist will have little difficulty when guided by them 
to fill up the representation by hand. 

A regular fortification, which we will suppose to have eight 
bastions, will afford another example. 

A person not conversant in such a subject, is in general puz- 
zled with plans and sections, and has very little idea of what is 
meant to be conveyed. 

But he would easily understand it if he should see every 
thing exhibited in a correct picture, especially where he has the 
the view of his object varied, as in a fortification, such as has 
been proposed. Let an isometrical ellipse be drawn expressing 
the internal circumference of the place ; and another concentric 
one, which marks the salient angles of the fortification on the 
principles already explained. Draw other guiding lines to every 
necessary point ; the lines of the fortification may be easily 
transferred from a common plan to the isometrical by the help 
of the scale of concentric ellipses, described above, fig. 7, which 
will serve also to lay down the length of the bastions and cur- 
tains, &c. in whatever direction they lie. Find the elevations 
of every part on the isometrical scale ; and thus the body of the 
place, the ditches, counterscarp, covered way, glacis, ravellins, 
and all the outworks, will be represented to the eye as they ap- 
pear in reality, and in every varied position, with the advantage 
of having all the admeasurements laid down with geometrical 
precision. 

If the artist should think the vertical lines in such an exhibi- 
tion too small to give a correct idea of all the minute elevations, 
there would be no harm in his increasing the scale in that di- 
mension in any desired proportions. 

The face of a hilly or mountainous country, like Switzer- 
land, or the district of the lakes in the north part of England, 
will afford another example. 



ISOMETRICAL PERSPECTIVE. 193 

Isometrical horizontal lines may be drawn representing lines 
in the level from which the height of the mountains is to be 
reckoned, so that vertical lines drawn from the summits of the 
mountains may meet them, on which the heights may be marked 
(as well as recorded in figures, if required). And the moun- 
tains themselves may be drawn in their topographical situation. 
Their bearings may be marked by the help of the isometrical 
compass described in page 189. It would be easy to transfer 
them from a common map to the isometrical plan ; and thus 
the face of the country might be represented just as it would 
appear from the commanding height which the isometrical per- 
spective supposes. 

Yet, as the slopes of hills and mountains are seldom so steep 
as the line of sight, it might sometimes suit the purpose to re- 
present the height of elevations as twice or three times the 
reality, in order that mountains might project an outline on the 
plane behind ; otherwise, the summit might be projected on the 
mountain itself, which would, in a degree, destroy the effect of 
a picture. 

This art might be advantageously employed also for tracing 
what is below the surface of the earth, as well as what is above 
it. It may be applied to geological purposes, and give, not only 
the order of the strata, but their variations and their geographi- 
cal situations. And for this purpose it might be useful to in- 
crease the vertical scale, in a great proportion, above the hori- 
zontal. It would be easy to mark the dip, or rise of the strata, 
as well as of the earth above them : to represent their various 
disruptions, to show the situation and extent of fissures and me- 
tallic veins, to mark the boundaries where the upper strata have 
been swallowed up, or cease to appear, or where the under 
strata push up towards the day. It would be easy to mark the 
variations in the thickness of the strata in different places, and 
to record the result of experiments made at any point, by bor- 
ing or sinking shafts, which might be done by drawing a verti- 
cal line downward, so as to represent the thickness of the lami- 
nae, which might be marked by different colours. By such a 
method, the geologist might obtain a map of the country, which 
might exhibit, at one view, the general results of all the experi- 
ments and inquiries that had been made relative to that science, 
and the owner of an estate might record, in a small compass, all 
that is known respecting its minerals, and be able, from a com- 
prehensive view of them all, to judge of the probability of suc- 
cess in sinking a shaft, or driving a level. He might also make 
good use of this perspective in tracing his shafts and drifts in 
all their windings, elevations, and depressions, and comparing 
them with the surface above, marking also the veins and strata 

S 



194 ISOMETRICAL PERSPECTIVE. 

in which they run. For if the artist knows what is beneath the 
surface, he has no difficulty in representing it as transparent. 
He must be careful however not to perplex himself by lines too 
much multiplied, and take advantage of his being able to paint 
the lines with different colours, for the purposes of distinction ; 
and he must also use a considerable address in throwing out 
such lines as would be of little use, and retaining such as will 
produce the effect of a picture, which should be well preserved 
in order to make the exhibition easily intelligible. 

If he should wish to make a drawing of minerals or crystals, 
this perspective would be well suited to the purpose. 

The point, however, on which the writer of this paper can 
speak with the greatest confidence is on the representation of 
machines and philosophical instruments ; having been himself 
so much in the habit of practically applying to them the princi- 
ples that have been detailed : and this he has exemplified in 
the plates. 

The correct exhibition of objects would be much facilitated 
by the use of this perspective, even in the hands of a person 
who is but little acquainted with the art of drawing ; and the 
information given by such drawings is much more definite and 
precise than that obtained by the usual methods, and better fitted 
to direct a workman in execution.* 

* The author has transcribed this interesting paper from the first volume of the 
Transactions of the Cambridge Philosophical Society. The method is peculiarly de- 
serving of the attention, and, in many cases, the adoption, of mechanics and engi- 
neers. 

Some useful exemplifications of isoperimetrical perspective are given in the 18th 
and 19th volumes of the Mechanic's Magazine ; and while this sheet was going 
through the press, I learnt that Mr. JopUng is now printing a small treatise on the 
subject, for the use of mechanics and artificers. 



MENSURATION. 195 



CHAPTER VIII. 

MENSURATION OF SUPERFICIES AND SOLIDS. 

Section I. — Mensuration of Superficies. 



The following rules will serve to find the areas or superficial 
contents of the figures whose names respectively precede 
them. 

1. Rectangle, Square, Rhombus, or Rhomboid. Multiply 
the base into the height, for the area. 

2. Triangle. Half base into the height. Or, continual 
product of two sides, and half the natural sine of their inclined 
angle. 

Or, when three sides, as a b, a c, b c, are given, their half sum 
being s ; then 

area= </[s X (s — a b) X (s — Ac) X s — b c)]. 

3. Trapezium. Base into 5 sum of the perpendiculars. 

4. Trapezoid. Multiply half the sum of the parallel sides 
into the perpendicular distance between them. 

5. Irregular Polygon. Divide it into trapeziums, or trape- 
zoids, and triangles, and find their areas separately : their sum 
is the area of the polygon. 

6. Regular Polygon. Multiply the square of the side given 
into the proper " multiplier for areas," printed in the table in 
Prob. 15, Practical Geometry, the product will be the area. 

7. Circle, diameter : circumf :: 113 : 355 
or, diameter : circumf :: 1 : 3*141593 

circumf : diameter :: 1 : '318309 
area= diameter squared x '785398 
area= circumference squared x '079577 
area=i diameter x h circumf. 

8. Circular arc. Radius of the circle X '017453 x degrees 
in the arc = its length. 

9. Circular sector. \ arc x radius = area. 

10. Circular segment. Multiply the square of the radius 
by either half the difference of the arc (of the segment) and its 



196 MENSURATION OP SUPERFICIES. 

sine, or by half their sum, according as the segment is less or 
greater than a semicircle : the product will be the area. 

11. Parabola, § of the product of base and height = area. 

12. Ellipse. Transverse axis x conjugate axis x '785398 
= area. 

13. The side of a square whose area shall be equal to that of 
a given circle, is nearly | of the diameter, or more nearly -f-i. d, 
or || d, or -iff d, or iff d ; each approximating more nearly 
than the former. 



Section II. — Mensuration of Solids. 

1. Prism. 1. Superficies. Multiply the perimeter of one 
end by the length or height of the solid ; the product will be 
the surface of the sides. To this add the areas of the two ends : 
the sum is the whole surface. 

2. Solidity or Capacity = area of the base x the height. 
Note. The same rules serve for the surface and capacity of 

a cylinder. 

2. Pyramid, or Cone. 1. Surface=£ perimeter of the base 
X slant height. 

2. Capacity = area of base x \ height. 

3. Frustrum of Pyramid. 1. Surface = \ sum of the peri- 
meters of the two ends x slant height. 

2. Capacity. Add a diameter or a side of the greater base 
to one of the less ; from the square of the sum subtract the pro- 
duct of the said two diameters or sides : multiply the remain- 
der by a third of the height ; and this last product by '785398 
for circles, or by the proper multiplier for polygons ; the last 
product will be the capacity. 

That is, capacity=[(D + o?) 2 — n d~\\ m h. 

4. Sphere. 1. Surface— diameter squared x 3*141593. 
2. Capacity— diameter cubed x '5236 

or = circumference cubed x '016887. 

5. Spheric segment. 1. Surface= circumi. sphere x height 

of segment. 
2. Capacity = -5236 h* (3 d — 2 h) : where d = diam. 

h = height. 
= -5236 A 3 (3 r*+h 2 ) : where r=rad. of the 
segment's base. 
5. Paraboloid. Capacity = half base x height. 
This is a figure produced by the rotation of a parabola upon 
its axis. 



MENSURATION OP SOLIDS. 



197 



6. Sphe?*oid. — This is a solid generated by the revolution of 
an ellipse about one of its axes. To find its capacity multiply 
the square of the revolving axis by the fixed axis, and that pro- 
duct by -5236. 

7. Regular or platonic bodies, are comprehended by like, 
equal, and regular plane figures, and whose solid angles are all 
equal. 

There are only five regular solids, viz. 

The tetraedron, or regular triangular pyramid, having 4 tri- 
angular faces ; 

The hexaedron, or cube, having 6 square faces ; 

The octaedron, having 8 triangular faces ; 

The dodacaedron, having 12 pentagonal faces ; 

The icosaedron, having 20 triangular faces. 

Prob. 1. To find either the surface or the solid content of 
any of the regular bodies. — Multiply the f proper tabular area 
or surface (taken from the following table) by the square of the 
linear edge of the solid for the superficies. And 

Multiply the tabular solidity in the last column of the table 
by the cube of the linear edge for the solid content. 



Surfaces and Solidities of regular Bodies, the side being 
unity or 1. 



No. of 

sides. 


Name. 


Surface. 


Solidity. 


4 

6 

8 

12 

20 


Tetraedron 

Hexaedron 

Octaedron 

Dodecaedron 

Icosaedron 


1-7320508 
6-0000000 
3-4641016 
20-6457288 
8-6602540 


0-1178513 
1-0000000 
0-4714045 
7-6631189 
2-1816950 



2. The diameter of a sphere being given to find the side of 
any of the platonic bodies, that may be either inscribed in the 
sphere, or circumscribed about the sphere, or that is equal to 
the sphere. 

Multiply the given diameter of the sphere by the proper or 
corresponding number, in the following table, answering to the 
thing sought, and the product will be the side of the platonic 
body required. 

27 s2 



198 



MENSURATION OF SOLIDS. 



The diam. of a 


That may be 


That may be cir- 


That is equal 


sphere being 1 ; 


inscribed in 


cumscribed about 


to the sphere, 


the side of a 


the sphere, is 


the sphere, is 


is 


Tetraedron 


0-8164966 


2.4494897 


1-6439480 


Hexaedron 


0-5773503 


1-0000000 


0-8059958 


Octaedron 


0-7071068 


1-2247447 


1-0356300 


Dodecaedron 


0-3568221 


0-4490279 


0-4088190 


Icosaedron 


0-5257309 


0-6615845 


0-6214433 



3. The side of any of the five platonic bodies being given, to 
find the diameter of a sphere, that may either be inscribed in 
that body, or circumscribed about it, or that is equal to it. As 
the respective number in the table above, under the title in- 
scribed, circumscribed, or equal, is to 1, so is the side of the 
given platonic body to the diameter of its inscribed, circum- 
scribed, or equal sphere. 

4. The side of any one of the five platonic bodies being given 
to find the side of the other four bodies, that may be equal in 
solidity to that of the given body. — As the number under the 
title equal in the last column of the table above, against the 
given platonic body, is to the number under the same title, 
against the body whose side is sought, so is the side of the given 
platonic body to the side of the body sought. 

Besides these there are thirteen demiregular bodies, called 
Solids of Archimedes. They are described in the Supplement 
to Lidonne's Tables de Tous les Diviseurs des Nombres, &c. 
Paris, 1808 ; twelve of them were described by Abraham Sharp, 
in his Treatise on Polyedra. 



Section III. — Approximate Rules. 

1. When the area of a field is known in square yards, to re- 
duce them to acres, instead of dividing by 4840, multiply by 
•0002^., which is much easier. 

Thus, suppose the area is found to be 56870 yards. 

Then 56870 

•0002 



Take^ 
Or*of T V 

The sum is 



of this, 
. . . viz. 



11-3740 
3791 



11-7531 acres: the true answer 
[is 1 1 -75 acres. 



APPROXIMATE RULES IN MENSURATION. 199 

2. For regular Polygons. — Let s be the side, then 
area of trigon =§ s3 +to ^ 

pentagon = s 3 + T V s 2 — T |^ s 3 
hexagon =2 s 3 +to s 3 
heptagon = T |^ °f square of 11 s 

octagon = T V of square of 7 s — 

5 

nonagon = 6 s 3 + T \ s 3 — T ^ s 3 

decagon =7 s 3 + T 7 ^ s 2 

undecagon = 9 s 3 + T 4o of 9 s 2 

dodecagon =11 s 3 +y s 3 . 

Where a table of polygons is at hand, it is best to employ it. 
In other cases, one or other of these approximations may occur 
to a well exercised memory. 

Ex. 1. The side of a pentagon is 20. Required the area. 
S 3 =400 
_8_s 3 =320 8 times A s 3 



From their sum = 720 

Take T i 7 s 9 = 32 T Vof T 8 oS 3 

Area required = 688 : the true area is 688*19. 



Ex. 2. The side of an octagon is 20. Required the area. 
Square of 7 s= 19600= square of 140 

From T V of ditto = 1960 

Take !± = 28 

5 



Area of the octagon = 1932 : the true area is 1931*37. 



Ex. 3. The side of a nonagon is 20. Required the area. 
s 3 =400, 6 s 2 = 2400 
Add ft s 2 = 80 



From the sum 2480 
Takers* = 8 = T Vof T 2 o 



Area of the nonagon = 2472 : the true area is 2472*72. 



200 APPROXIMATE RULES IN ARITHMETIC. 

Ex. 4. The side of a dodecagon is 20. Required the area. 
s 2 = 400, 11 s 2 = 4400 

Add ls a = 80 



Area of , the dodecagon = 4480 : the true area is 4478'46. 

3. Length of circular arc. From 8 times the chord of half 
the arc, subtract the chord of the whole arc, and § of the re- 
mainder will be the length of the arc, nearly. 

Note. — The chord of half the arc is equal to the square 
root of the sum of the squares of the versed sine or height, and 
half the chord of the entire arc. 

Or, apply a fine flexible string to the arc, then stretch it out 
straight, and measure it. 

4. tdrc of a quadrant nearly equal 2\ time the chord of the 
quadrant. 

This is true within about the 4000th part. 

5. Periphery of an ellipse. — Multiply the square root of 
half the sum of the squares of the two axes by 3*141593, and 
the product will be the periphery, nearly. 

Diminish this by its 200th part, and the result will be still 
more correct. 

6. Jirea of a circle = nearly to \\ of the square of the dia- 
meter. Or, multiply d 3 by 11, and divide by 2 and by 7. 

This is true within the 2500th part ; or, add to Id 3 t ^ of 
J d 2 ; the sum will be the area true to within the 5000th 
part. 

Area of a circle = nearly to ¥ 7 ¥ of the square of the circum- 
ference. Or, multiply c 2 by 7, and divide by 8 and 11. 
True within the 2600th part. 

7. Circular segment. — To the chord of the whole arc add 
-f of the chord of half the arc ; multiply the sum by the versed 
sine or height of the segment ; and T \ of the product will be 
the area nearly. 

Or, (c+f </i c*+v 2 ) T \ v = area. 

8. To find the content of irregular plane figures from 
an accurate plan. 

If the plan be not upon paper, or fine drawing pasteboard 
of uniform texture, let it be transferred upon such. Then cut 
out the figure separately close upon its boundaries : and cut out 
from the same paper a square of known dimensions according 
to the scale employed in drawing the plan. Weigh the two 
separately in an accurate balance, and the ratio of the weights 
will be the same as that of the superficial contents. 



APPROXIMATE RULES I MENSURATION. 201 

If great accuracy be required, cut the plan into 4 portions, 
called 1, 2, 3, 4. First, weigh 1 and 2 together, 3 and 4 to- 
gether, and take their sum. Then weigh 1 and 3 together, 2 
and 4 together, and take their sum. Lastly, weigh 1 and 4 to- 
gether, 2 and 3 together, and take their sum. The mean of the 
four aggregate weights thus obtained, compared with the weight 
of the standard square, will give the ratio of their surfaces very 
nearly. 

*^* I have employed in this operation a balance which turns 
with the 100th part of a grain. The results are proportionally 
accurate. 

9. Prism. — l = length, b— breadth, D=depth, all in inches : 
then T 4 o 9 o"o o l b d = content in yards, nearly. 

If l, b, d, be in feet, as suppose the dimensions of a corn bin ; 
then '8 l b d = the content in Winchester bushels. This is 
about one bushel in 200 in defect. 

Ex. Suppose l = 125 inches, b = 25, d = 24. 
Then 125x25=1-2X^ = 3125 ; 
and 3125X24=6250X12=75000 

T4Aoo°f 75000 
9 



2 ) 675000 
7000 ) 337500 



4-8214 yards : the true answer is 4*8225. 

For wrought iron square bars allow 100 inches in length of 
an inch square bar to a quarter of a cwt. in weight ; and so in 
proportion. This is easily remembered, because the word hun- 
dred occurs twice. 

An inch square cast iron bar would require 9 feet in length 
for a quarter of a cwt. 

Or, take T l T of the product of the breadth and thickness, 
each in eighths of an inch, the result is the weight of one foot 
in length, in avoirdupois pounds. 

Or, one foot in length of an inch square bar weighs 3-J- 
pounds. 

Bricks of the usual size require 384 to a cubic yard. A rod 
of brick work, brick and a half thick, requires 4356 bricks. 

10. Cylinder. One tenth diameter squared, ( T V^ a ? d being 
taken in inches,) gives the content in ale-gallons of a yard in 
length. 



202 APPROXIMATE RULES ! MENSURATION. 

This rule gives a result defective only by the 376th part. 

I d 2 x -00283257 = imperial gallons in a cylinder, di- 
mensions in inches. 

11. Timber measuring. — Let l denote the length of a tree 
in feet and decimals, and g the mean girth, taken in inches : then 
the following rules given by Mr. Andrews may be employed. 

Rule 1. — No allowance for bark. 

IG 2 . L G 2 

22qI == cu ^ c f eet > customary, and ztttz = cubic feet, true content. 
Rule 2.— To allow fth for bark. 

I6 J I G 2 

^rr— = cubic feet, customary, and <^^= cubic feet, true content 

Rule 3.— To allow ith for bark. 

i e s l g 2 

ogig = cubic feet, customary, and rr^r = cubic feet, true content. 

Rule 4.— To allow JLth for bark, 
i 2 

I G 2 I 6 2 

2Y42 = cubic feet, customary, and ^TTTv = cubic feet, true content. 

Example by Rule 1. — No allowance for bark. 
A tree 40 feet long, and 60 inches whole girth or circumference. 

40 X 60 2 40 X 60 2 

— — — — = 62£ cubic ft. customary, and = 79| cubic feet, true content 

Ex. by Rule 2. — A tree feet 50 long, and 49 inches circumference. 
50 X 49 2 50 x 49 2 

= 40 cubic ft. customary, and ^ — = 50| cubic feet, true content. 

If g as well as x be in feet, then *08 x g 2 = content, nearly. 

%* When the difference between the girths at the two ends 
is considerable, it is best to find the content of the tree as though 
it were a conic frustrum, and make the usual allowances after- 
wards. 

12. Sphere ~\ of the cube of the diameter = capacity. Use 
the component factors, 3 and 7, in dividing by 21. 

This rule gives a result true to its 2600th part. 
Or, multiply the cube of the circumference by -0169, for the 
capacity. 

13. To find the capacity, or solid content of an irregular 
body. — Procure a prismatic or cylindric vessel that will hold it. 
Put in the body, and then pour in water to cover it, marking 
the height to which the water reaches. Then take out the 
body, and observe accurately how much the water has de- 
scended in consequence. The capacity of the prism or cylinder 
thus left dry by the water will be evidently equal to that of the 
body. 

If the vessel will not hold water, sand may be employed, 
though not with quite so much accuracy. 



APPROXIMATE RULES I MENSURATION. 203 

In this manner, too, a -portion of a body may be measured 
without detaching it from the rest, by simply immersing that 
portion. 

Even an irregular vessel may be employed for this purpose. 
In which case it should be placed in a larger vessel, and then 
filled with water. Then submerge the body whose magnitude 
you wish to determine, the quantity of water that has run over, 
and is caught in the exterior vessel, will be the measure. It 
may be weighed, and its cubic measure estimated by allowing 
1000 avoirdupois ounces to the cubic foot. (See farther, Hy- 
drostatics and Specific Gravity.) 

14. To find the contents of surfaces and solids not re- 
ducible to any known figure, by the equidistant ordinate 
method. — The general rule is included in this proposition : 
viz. — If any right line a n be divided into any even number of 
equal parts, Ac, c e, e g, &c. and at the points of division be 
erected perpendicular ordinates a b, c d, e f, &c. terminated 
by any curve b h o : then, if a be put 
for the sum of the first and last ordi- 
nates, a b, n o, b for the sum of the 
even ordinates, c d, g h, l m, &c. viz. 
the second, fourth, sixth, &c. and c for 
the sum of all the rest, e f, i k, &c. 



viz. the third, fifth, &c. or the odd A c e g i l n 
ordinates, excepting the first and last : then, the common dis- 
tance ac, c e, &c. of the ordinates being multiplied into the 
sum arising from the addition of a, four times b, and twice 
c, one third of the product will be the area a b o n, very 
nearly. 

™, , . A + 4 b + 2 c ". e 

lhat is, . d=»= area, d being =a c=c e, &c. 

The same theorem will equally serve for the contents of all 
solids, by using the sections perpendicular to the axis instead 
of the ordinates. The proposition is quite accurate, for all 
parabolic and right lined areas, as well as for all solids gene- 
rated by the revolutions of conic sections or right lines about 
axes, and for pyramids and their frustrums. For other areas 
and solidities it is an excellent approximation. 

The greater the number of ordinates, or of sections, that 
are taken, the more accurately will the area or the capa- 
city be determined. But in a great majority of cases five 
equidistant ordinates, or sections, will lead to a very accurate 
result. 

In cask-guaging, indeed, three sections will be usually 
sufficient. Thus, taking the bung and head diameters, and 



204 APPROXIMATE RULES I MENSURATION. 

a diameter mid-way between them ; the sum of the squares of 
the bung and head diameters, and of the square of double the 
middle diameter, multiplied into the length of the cask, and 
then into '785398, will give six times the content of the cask, 
very nearly. Or, if h, b, and m, represent the head, bung 
and middle diameters respectively, and l the length, all in 
inches ; then (h 3 -f 4 m 3 + b 3 ) X l X *1309=content of the 
cask in inches. 

A similar method may be advantageously adopted in all 
cases of ullaging either standing or lying casks, by taking the 
areas at the top, bottom, and middle, of the liquor. See Hut- 
ton's Mensuration, part iv. 

Example. The bung diameter of a cask is 32 inches, the 
head diameter 24, the middle 30*2, the length 40. Required 
the content in ale gallons of 282 cubic inches, and in wine gal- 
lons of 231. 

The former multiplier, divided by 282 and 231 respectively, 
gives -0004 T 7 T and -0005f , for the proper multipliers. 

Hence (32 2 -f24 3 -f4 . 30'2 3 ) X 40 x -0004 T 7 T =97-44 ale gal- 
lons : 

And (32 3 +24 2 +4 . 30-2 3 ) X 40 X -00051=118-95 wine gal- 
lons : 

Or y of 97*44 (the ale gallons) gives 119, the wine gallons, 
very nearly. 

Also |- of 118-95 (the wine gallons) =99*125 imperial gal- 
lons. See the table of factors, p. 19. 



MECHANICS : STATICS. 205 



MECHANICS. 

1. Mechanics is the science of equilibrium and of motion. 

2. Every cause which tends to move a body, or to stop it 
when in motion, or to change the direction of its motion, is 
called a force or power. 

3. When the forces that act upon a body, destroy or annihi- 
late each other's operation, so that the body remains quiescent, 
there is said to be an equilibrium. 

4. Statics has for its object the equilibrium of forces applied 
to solid bodies. 

5. Dynamics relates to the circumstances of the motion of 
solid bodies. 

6. Hydrostatics is devoted to the equilibrium of fluids. 

7. Hydrodynamics relates to their motion, and connected 
circumstances. 

8. The properties and operation of elastic fluids are often 
treated distinctly, under the head of Pneumatics. 

9. A single force, which would give to a physical point, or 
to a body, the same motion both in velocity and direction as 
several forces acting simultaneously, is called the resultant of 
those forces, while they are called the constituents or the com- 
posants of the single resulting force. 

10. The action of a force is the same in which ever point of 
its direction it is applied ; unless the manner of its action be 
changed. 

11. Vis inertias, or power of inactivity, is defined by New- 
ton to be a power implanted in all matter, by which it 
resists any change attempted to be made in its state, that is, 
by which it requires force to alter its state, either of rest or 
motion. 



28 



206 



MECHANICS : STATICS. 



CHAPTER IX. 



STATICS. 



Section I. — Parallelogram of Forces. 



1. The resultant of any two forces whatever, which act 
upon a physical point, and which are represented by lines 
taken in their respective directions from that point, has for its 
magnitude and direction the diagonal of the parallelogram con- 
structed upon those forces. 

2. Two forces and their resultant may each be represented 
by the sine of the angle formed by the directions of the two 
others. 

3. If f, f, represent two forces, and a the angle made by 
their respective directions ; also, if r be the resultant, and a 
the angle which it makes with the direction of the force f : 
then 

R=v/(f 2 +/ 3 ± 2 f/cos a) (1) 

f sin a 



tan a = 



f+ f cos A 



(2) 



4. These propositions are of extensive, nay, of almost univer- 
sal application, in the constructions of architects, mechanics, 
and civil engineers. Two simple examples may here suffice : 
others will occur as we proceed. 

Ex. 1. Suppose that a weight b is at- 
tached by a stirrup to the foot of a king- 
post a b, which is attached to two rafters 
A c, A d, in the respective positions shown 
in the marginal diagram. Then if a e be 
set off upon a b, in numerical value equal 
to the vertical strain upon a b, and the 
parallelogram a f e g be completed, a f 
measured upon the same scale will show 
the strain upon the rafter a c, and a g the strain upon A D. 




MECHANICS : STATICS. 



207 




- — jr.. 
E 



Ex. 2. Let it be proposed to compare the strains upon the 
tie-beams a d, and the struts a c, when they sustain equal 
weights b, in the two different posi- 
tions indicated by the figures. Let 
ae in one figure, be equal to the 
corresponding vertical line ae in 
the other, and in each represent the 
numerical value of the weight b, 
that hangs from a. Through e in 
both figures, draw lines parallel to 
d a, a c, respectively, and let them 
meet a c, and d a produced in f and 
g : then a p e g in each figure is the 
parallelogram of forces by which 
the several strains are to be measur- 
ed, a G represents the tension upon 
the tie-beam a d, and a f the strain 
upon the strut a c. Both these lines 
are evidently shorter in the lower 
figure than they are in the upper, 
a e being of the same length in both : therefore the first figure 
exhibits the most disadvantageous position of the beams. It is 
evident also, that while c a tends upwards and d a downwards, 
the greater the angle d a c, the less is its supplement c a g, and 
the less the sides f a, a g, of the parallelogram. 

The truth of the general proposition relative to the parallel- 
ogram of forces admits of obvious experimental confirma- 
tion by means of two spring steelyards. Let any known 
weight, as 10, 15, &c. pounds, represented by b in the figure 
to Ex. 1, hang from a cord e b, and from a knot at e let 
two other cords, e f, e g, proceed, and hang from spring 
steelyards at f and g : then it will be found universally that 
the weights sustained by those steelyards, will be to the 
weight b hanging vertically, as the respective sides e f and 
e G of the parallelogram to its diagonal e a. It will, 
hence, be easy to exhibit those relations in all desirable va- 
rieties. 

5. Three forces not situated in the same plane, but applied 
to the same point, have a resultant represented in magnitude 
and direction by the diagonal of the parallelopiped con- 
structed upon the parts of the directions of those forces which 
represent their respective magnitudes, and drawn from their 
point of application. 

6. We may always decompose or resolve a force into three 
others, parallel respectively to three given lines. Each com- 
posant may be found by multiplying the force which we 




208 MECHANICS : STATICS. 

would decompose by the cosine of the angle which its direction 
makes with the axis to which that composant is parallel. 

7. If any number of forces be kept in equilibrio by their 
mutual actions ; they may all be reduced to two equal and 
opposite ones. — For, any two of the forces may be reduced 
to one force acting in the same plane ; then this last force 
and another may likewise be reduced to another force acting 
in their plane : and so on, till at last they be all reduced to the 
action of only two opposite forces ; which will be equal, as 
well as opposite, because the whole are in equilibrio by the sup- 
position. 

8. Prop. To find the resultant of several forces concurring in 
one point, and acting in one plane. 

1st. Graphically. Let, for example, four forces, a, b, c, d, 
act upon the point p, in magnitudes and directions represented 
by the lines p a, p b, p c, p d. d 

From the point a draw a b parallel 
and equal to p b ; from b draw b c paral- 
lel and equal to p c ; from c draw c d pa- 
rallel and equal to p d ; and so on, till all 
the forces have thus been brought into 
the construction. Then join p d, which 
will represent both the magnitude and the direction of the re- 
quired resultant. 

This is, in effect, the same thing as finding the resultant of 
two of the forces a and b ; then blending that resultant with a 
third force c ; their resultant with a fourth force d ; and so on. 
A careful construction upon a well chosen scale will give a re- 
sultant true to within its 250th part. 

2d. By computation. Drawing the lines a a, a b', &c. re- 
spectively parallel and perpendicular to the last force p d ; we 
have 
d 6=a a -\- bb' -f- cc'—a sin a p d -f- b sin b p d 4- c sin c p d 

P 5 = Pa + a/34-/3y4-y8 = A COS APD + BCOSB P D + C COS CP D + D 

dS 
tan d p5=— p c?= ^/ (p 8 2 +d6 2 )=r 8 sec d p 3. 

The numerical computation is best effected by means of a table 
of natural sines. 

9. The resultant of two parallel forces acting in the same way, 
is equal to their sum ; and the distances of the direction of that 
resultant from those of the composants are reciprocally propor- 
tional to those composants. 

10. The resultant of several parallel forces, whether confined 
to one plane or not, is equal to the sum of those forces, giving 
to the forces which act in one sense the sign +, and to those 
which act contrarily the sign — . 



STATICS : CENTRE OP GRAVITY. 209 

Section II. — Centre of Gravity. 

1. Gravity is the force in virtue of which bodies left to them- 
selves fall to the earth in directions perpendicular to its surface. 

2. We may distinguish between the effect of gravity and 
that of weight, by observing that the former is the power of 
transmitting to every particle of matter a certain velocity 
which is absolutely independent on the number of material 
particles ; while the latter is the effort which must be exer- 
cised to prevent a given mass from obeying the law of gravity. 
Weight, therefore, depends upon the mass ; but gravity has 
no dependence at all upon it. 

3. The centre of gravity of any body or system of bodies is 
that point about which the body or system, acted upon only 
by the force of gravity, will balance itself in all positions : or 
it is a point which, when supported, the body or system will 
be supported, however it may be situated in other respects. 

The centre of gravity of a body is not always within the 
body itself : thus the centre of gravity of a ring is not in the 
substance of the ring, but in the axis of its circumscribing cy- 
linder ; and the centre of gravity of a hollow staff, or of a bone, 
is not in the matter of which it is constituted, but somewhere 
in its imaginary axis ; every body, however, has a centre of 
gravity, and so has every system of bodies. 

4. Varying the position of the body will not cause any 
change in the centre of gravity ; since any such mutation will 
be nothing more than changing the directions of the forces, 
without their ceasing to be parallel ; and if the forces do not 
continue the same, in consequence of the body being supposed 
at different distances from the earth, still the forces upon all 
the moleculae vary proportionally, and their centre remains 
unchanged. 

5. When a heavy body is suspended by any other point 
than its centre of gravity, it will not rest unless that centre is 
in the same vertical line with the point of suspension : for in 
all other positions the force which is intended to ensure the 
equilibrium will not be directly opposite to the resultant of 
the parallel forces of gravity upon the several particles of the 
body, and of course the equilibrium will not be obtained. 
(See art. 9, on Pendulums.) 

6. If a heavy body be sustained by two or more forces, their 
directions must meet, either at the centre of gravity of that 
body, or in the vertical line which passes through it. 

7. When a body stands upon a plane, if a vertical line 
passing through the centre of gravity fall within the base on 
which the body stands, it will not fall over ; but if that verti- 

t 2 



210 



STATICS : CENTRE OF GRAVITY. 




cal line passes without the base, the body will fall, unless it be 
prevented by a prop or a cord. When the vertical line falls 
upon the extremity of the base, the body may stand, but the 
equilibrium may be disturbed by a very trifling force ; and 
the nearer this line passes to any edge of the base the more 
easily may the body be thrown over ; the nearer it falls to the 
middle of the base, the more firmly the body stands. 

Upon this principle it is that leaning towers have been built 
at Pisa, and various other places ; the vertical line of direction 
from the centre of gra- 
vity falling within the 
base. And, from the 
same principle it may be 
seen that a wagon loaded 
with heavy materials, as 
b, may stand with perfect 
safety, on the side of a 

convex road, the vertical line from the centre of gravity fall- 
ing between the wheels ; while a wagon a with a high load, 
as of hay, or of wool-packs, shall fall over, because the vertical 
line of direction falls without the wheels. 

8. To find the centre of gravity mechanically, it is only 
requisite to dispose the body successively, in two positions of 
equilibrium, by the aid of two forces in vertical directions, ap- 
plied in succession to two different points of the body ; the point 
of intersection of these two directions will show the centre. 

This may be exemplified by particularizing a few methods. 
If the body have plane sides, as a piece of 
board, hang it up by any point, then a plumb- 
line suspended from the same point will pass 
through the centre of gravity ; therefore 
mark that line upon it : and after suspending 
the body by another point, apply the plum- 
met to find another such line ; then will their 
intersection show the centre of gravity. 

Or thus : hang the body by two strings from 
the same point fixed to different parts of the 
body ; then a plummet hung from the same 
tack will fall on the centre of gravity. 

Another method : Lay the body on the edge 
of a triangular prism, or such like, moving it to 
and fro till the parts on both sides are in equi- 
librio, and mark a line upon it close by the 
edge of the prism : balance it again in another 
position, and mark the fresh line by the edge 
of the prism ; the vertical line passing through 




STATICS I CENTRE OF GRAVITY. 211 

the intersection of these lines, will likewise pass through the 
centre of gravity. The same thing may be effected by laying 
the body on a table, till it is just ready to fall off, and then 
marking a line upon it by the edge of the table : this done in 
two positions of the body will in like manner point out the 
position of the centre of gravity. 

When it is proposed to find the centre of gravity of the 
arch of a bridge, or any other structure, let it be laid down 
accurately to scale upon pasteboard ; and the figure being 
carefully cut out, its centre of gravity may be ascertained by 
the preceding process. 

9. If on any plane passing through the centre of gravity of 
a body, perpendiculars be let fall from each of its molecular, 
the sum of all the perpendicular distances on one side of the 
plane will be equal to the sum of all those on the other side. 
And a similar property obtains with regard to the common 
centre of gravity of a system of bodies. 

10. The position, distance, and motion of the centre of gra- 
vity of any body is a medium of the positions, distances, and 
motions of all the particles in the body. 

11. The common centre of gravity, or of position, of two 
bodies divides the right line drawn between the respective 
centres of the two bodies in the inverse ratio of their masses. 

12. The centre of gravity of three or more bodies may, 
hence, be found, by considering the first and second as a single 
body equal to their sum and placed in their common centre of 
gravity, determining the centre of gravity of this imaginary 
body, and a third. These three again being conceived united 
at their common centre, we may proceed, in like manner, to a 
fourth ; and so on, ad libitum. 

Or, if b, b', b", &c. denote the masses of any bodies, 

d, d', d", &c. the perpendicular distances of their respective 

centres of gravity from any line or plane : then, the distance, 

A, of their common centre of gravity from any line or plane, 

• r ■ j x. *u- ±u -a B d+b' d'+b" d" &c. 

is found by this theorem : viz. A = ' — 

b -f b + b &c. 

13. If the particles or bodies of any system be moving 
uniformly and rectilineally, with any velocities and directions 
whatever, the centre of gravity is either at rest, or moves 
uniformly in a right line. 

Hence if a rotatory motion be given to a body and it be 
then left to move freely, the axis of rotation will pass through 
the centre of gravity : for, that centre, either remaining at rest 
or moving uniformly forward in a right line, has no rotation. 

Here too it may be remarked, that a force applied at the 
centre of gravity of a body, cannot produce a rotatory mo- 
tion. 




212 STATICS I CENTRE OP GRAVITY. 

14. The centre of gravity of a right line, or of a parallelo- 
gram, prism, or cylinder, is in its middle point ; as is also 
that of a circle, or of its circumference, or of a sphere, or of 
a regular polygon ; the centre of gravity of a triangle is some- 
where in a line drawn from an angle to the middle of the 
opposite side ; that of an ellipse, a parabola, a cone, a conoid, 
a spheroid, &c. somewhere in its axis. And the same of all 
symmetrical figures. 

15. The centre of gravity of a triangle is the point of in- 
tersection of lines drawn from the three angles to the middles 
of the sides respectively opposite : it divides each of those 
lines into two portions in the ratio of 2 to 1. 

16. In a Trapezium. Divide the figure into 
two triangles by the diagonal a c, and find the 
centres of gravity e and r of these triangles ; 
join e f, and find the common centre g of 
these two by this proportion, abc:adc : : f g : 
eg, orABCDiADC : : e f : e g. Or, divide 
the figure into two triangles by a diagonal b d . 

find their centres of gravity ; the line which joins them will 
intersect e f in g, the centre of gravity of the trapezium. 

17. In like manner, for any other plane figure, whatever 
be the number of sides, divide it into several triangles, and 
find the centre of gravity of each ; then connect two centres 
together, and find their common centre as above ; then con- 
nect this and the centre of a third, and find the common 
centre of these ; and so on, always connecting the last found 
common centre to another centre, till the whole are included 
in this process ; so shall the last common centre be that which 
is required. 

18. The centre of gravity of a circular arc is distant from 
the centre a fourth proportional to the arc, the radius, and the 
chord of the arc. 

19. In a circular sector, the distance from the centre of 

2 C ¥ 

the circle is ; where a denotes the arc, c, its chord, and 

3a' ' ' ' 

r the radius. 

20. The centres of gravity of the surface of a cylinder, of 
a cone, and of a conic frustrum, are respectively at the same 
distances from the origin as are the centres of gravity, of the 
parallelogram, triangle, and trapezoid, which are vertical sec- 
tions of the respective solids. 

21. The centre of gravity of the surface of a spheric seg- 
ment is at the middle of its versed sine or height. 

22. The centre of gravity of the convex surface of a sphe- 
rical zone, is in the middle of that portion of the axis of the 
sphere which is intercepted by the two bases of the zone. 



STATICS I CENTRE OF GRAVITY. 



213 



23. In a cone, as well as any other pyramid, the distance 
from the vertex is I of the axis. 

24. In a conic frustum, the distance on the axis from the 

3 R a 4- 2 R V -f- T 2 I 

centre of the less end, is \ h . — ■ '- where h the 

4 r 3 + xr + r 2 

height, r and r the radii of the greater and less ends. 

25. The same theorem will serve for the frustum Of any- 
regular pyramid, taking r and r for the sides of the two ends. 

26. In the paraboloid, the distance from the vertex is § axis. 
21. In the frustum of the paraboloid, the distance on the 

2 R 3 + r 3 : 
axis from the centre of the less end, is \ h . — — — - — where 

6 R 2 +r* 

h the height, r and r the radii of the greater and less ends. 

*^* Many other results are given in the first volume of my 
Mechanics. The preceding are selected as the most useful. 

The centre of gravity of the human body is always near 
the same place, viz. in the pelvis, between the hips, the ossa 
pubis, and the low- 
er part of the back- 
bone. Elevating the 
arms or the legs will 
elevate the centre of 
gravity a little : still, 
it is always so placed 
that the limbs may 
move freely round it, 
and this centre moves 
much less than if it 
were in any other 
part of the body. If 
a man walked upon 
wooden legs,the cen- 
tre of gravity of his 
body would describe 
portions of circles, as 
a b. If a man with two wooden legs were to run, the centre 
of gravity would describe portions of parabolas, as c d. But 
the flexibility of the joints and muscles of the human legs 
serves to take away the angles from these curves, and give a 
softer undulation, as e f. 

The centre of gravity of a human body, is not precisely in 
the same place as that of the statue of a man : for, in the for- 
mer, the substance is not, throughout, of the same density, in the 
latter it is. 
29 




214 STATICS : MECHANICAL POWERS. 



Section III. — Mechanical Powers, 

1. The simple machines of which the more complex ma- 
chines are constituted, and which, indeed, are often employed 
separately, are called Mechanical Powers, 

2. Of these we usually reckon six : viz. the lever, the wheel 
and axle, the pulley, the inclined plane, the wedge, and the 
screw. To these, however, is sometimes added the funicular 
machine, being that which is formed by the action of several 
powers, at different points of a flexible cord. 

3. Weight and Power, when regarded as opposed to each 
other, signify the body to be moved, or the resistance to be 
overcome, and the body of force by which that is accomplished. 
They are usually represented by their initial letters, w and p. 

Levers. 

1. A lever is an inflexible bar, whether straight or bent, 
and supposed capable of turning upon a fixed, unyielding point 



called a fulcrum 

2. When the fulcrum is between the p ^ —f 

power and the weight, the lever is said 
to be of the first kind. 

When the weight is between the 
power and the prop, the lever is of the 
second kind. 



W wfj 



or 



When the power is between the weight 
and the prop, or fulcrum, the lever is of 
the third kind. 

The hammer lever, or the operation 
of a hammer in drawing a nail, is some- 
times considered as a fourth kind. 

3. In all these cases when there is an equilibrium, it is 
indicated by this general property, that the product of the 
weight into the distance at which it acts, is equal to the pro- 
duct of the power into the distance at which it acts : the dis- 
tances being estimated in directions perpendicular to those 
in which the weight and power act respectively. Thus, in each 
of the three preceding figures, 

p . a f=w . B F, 
or the power and weight are reciprocally as the distances at 
which they act. 

And if, in the first figure, for example, the arm a p were 



STATICS : MECHANICAL POWERS. 



21, 



4 times pb,4 lbs. hanging at b would be balanced by 1 lb. at 
a. If a f were 5 times fb, 1 lb. at a would balance 5 lbs. at 
b ; and so on. 

4. If several weights hang upon a lever, some on one side 
of the fulcrum, some on the other, then there will be an equi- 
librium, when the sum of the products of the weights into 
their respective distances on one side, is equal to the several 
products of weights and distances on the other side. 

5. When the weight of the lever is to be taken into the 
account, proceed just as though it were a separate weight sus- 
pended at its centre of gravity. 

6. When two, three, or more levers act one upon another 
in succession, then the entire mechanical advantage which 
they supply, is found by taking, not the sum, but the pro- 
duct of their separate advantages. Thus, if the arms of three 
levers, acting thus in connexion, are as 3 to 1, 4 to 1, and 5 
to 1, then the joint advantage is that of 3x4x5 to 1, or 
60 to 1 : so that 1 lb. would, through their intervention, 
balance 60. 

7. In the first kind of lever the pressure upon the fulcrum 
= p+w : in the other two it is= p ^ w. 

8. Upon the foregoing principles depends the nature of 
scales and beams for weighing all bodies. For, if the dis- 
tances be equal, then will the weights be equal also ; which 
gives the construction of the common scales. And the Ro- 
man statera, or steel-yard, is also a lever, but of unequal 
arms or distances, so contrived that one weight only may serve 
to weigh a great many, by sliding it backwards and forwards 
to different distances upon the longer arm of the lever. In 
the common balance, or scales, if the weight of an article when 
ascertained in one scale is not the same as its weight in the 
other, the square root of the product of those two weights 
will give the true weight. 

9. From numerous examples of the power and use of the 
lever, one which shows its manner of application in the print- 
ing-presses of the late Earl Stanhope may be advantageously 
introduced. 

In the adjoining figure, let 
A b c d be the general frame of the 
press, connected by the cross 
pieces n o, d c. e is a centre con- 
nected with the frame by the bars 
en,er,eo. To this centre are 
affixed a bar k d, and a lever e f, 
to which the hand is applied 
when the press is used. 




216 



STATICS I MECHANICAL POWERS. 



There are also several other pieces connected by joints at 
n, g, i, k, l, m, o, h, which are so adjusted to each other that 
when the hand is applied to the lever e f at f, by pressing it 
downwards k l is brought into a horizontal line or parallel to 
g h or d c, in which situation n i g, o m h, also form each a 
straight line. It is evident that the nearer these different pieces, 
as above mentioned, are to a straight line the greater is the 
lever e f, in proportion to the perpendicular k s at the other 
end of the lever e k, formed by a perpendicular from k falling 
on f e produced. Consequently a small force applied at f will 
be sufficient to produce a very great effect at k, when i k, k e 
are nearly in a straight line, and so on, for the other pieces 
above mentioned. 

Hence the force applied by hand at f must be very con- 
siderable in forcing down g h, which slides on iron cylindrical 
bars, or in pressing any substance placed in the aperture p q, 
between the bar or plate and the frame d c. 

This contrivance is now often introduced into mechanism, 
under the name of the toggle, or of the knee joint. 

Wheel and Axle. 



1. The nature of this machine is suggested 
by its name. To it may be referred all 
turning or wheel-machines of different radii ; 
as well-rollers and handles, cranes, capstans, 
windlasses, &c. 



2. The mechanical property is the same as in 
the lever : viz. p . a c=w . b c : and the rea- 
son is evident, because the wheel and axle is 
only a kind of perpetual lever. 



3. When a series of wheels 
and axles act upon each other, 
so as to transmit and accumu- 
late a mechanical advantage, 
whether the communication be 
by means of cords and belts, or 
of teeth and pinions, the weight will be to the power, not 
as the sum, but as the continual product of the radii of 
the wheels to the continual product of the radii of the axles. 
Thus, if the radii of the axles, a, b, c, d, e, be each 3 inches, 
while the radii of the wheels, a, b, c, d, e, be 9, 6, 9, 10, and 
12 inches respectively : then w:p::9 x6 x9 XlOx 12: 





STATICS : MECHANICAL POWERS. 



217 



3x3x3x3x3:: 240 : 1. A computation, however, in which 
the effect of friction is disregarded. 

4. A train of wheels and pinions may also serve for the aug- 
mentation of velocities. Thus, in the preceding example, 
whatever motion be given to the circumference of the axle e, 
the rim of the wheel a will move 240 times as fast. 

And if a series of 6 wheels and axles, each having their dia- 
meters in the ratio of 10 to 1, were employed to accumulate 
velocity, the produced would be to the producing velocity, as 
10 6 to 1, that is, as 1000000 to 1. 



Pulley. 

1. A pulley is a small wheel, commonly made of wood or 
brass, which turns about an iron axis passing through the 
centre, and fixed in a block, by means of a cord passed round 
its circumference, which serves to draw up any weight. The 
pulley is either single, or combined together, to increase the 
power. It is also either fixed or moveable, according as it 
is fixed to one place, or moves up and down with the weight 
and power. 

2. If a power sustain a weight by means of a fixed pulley : 
the power and weight are equal. 



3. When a power sustains a weight by a 
system of moveable pulleys, each embraced 
by a cord attached on one part to a fixed point, 
and on the other to the centre of the pulley 
next above it, as in the margin : then if the 
cords are parallel to each other, each pulley 
gives a mechanical advantage of two to one ; 
and the whole system and advantage denoted 
by that power of 2 which is equal to the num- 
ber of pulleys. Here p : w :: 1 : 2 4 :: 1 : 16. 



4. When there are three, four, or any other number of pul- 
leys in a fixed block, and an equal number in a moveable block, 
capable of ascending and descending, the system is called a 
muffle ; and the weight is to the power as 1 to twice the num- 
ber of pulleys in each block. 

5. The friction of a system of pulleys, or even of a single 
pulley, is very great, according to the common mode of con- 
struction. But it may be reduced considerably by means of 

U 




218 



STATICS : MECHANICAL POWERS. 




Mr. Garnett's patent friction rollers, which 
produce a great saving of labour and ex- 
pense, as well as in the wear of the machine, 
both when applied to pulleys and to the 
axles of wheel carriages. His general prin- 
ciple is this : between the axle and nave, 
or centre pin and box, a hollow space is left, 
to be filled up by solid equal rollers nearly 
touching each other. These are furnished 
with axles inserted into a circular ring at each end, by which 
their relative distances are preserved ; and they are kept paral- 
lel by means of wires fastened to the rings between the rollers, 
and which are rivetted to them. The above contrivance is ex- 
hibited in the annexed figure. 

6. A useful combination of the wheel and axle, a fixed and 
a moveable pulley, is exhibited in the marginal diagram. 
The load, as of stones or 
bricks to build a wall, is 
raised from l to a, thus : a 
rope b p l is fixed at one end 
to a hook b, and passes over 
a pulley at p. That pulley, 
p, is drawn along horizontal- 
ly from p to f by means of a 
man who turns the handle h, 
and thus winds up the cord 
p p h upon the roller. As the distance p b lengthens, the por- 
tion p l shortens ; and the length of rope is so adjusted, that 
when the pulley p is brought to be above a, the basket l has 
reached that place. 




Inclined Plane. 



1. A body which touches a plane only in one point, can only 
remain in equilibrio so long as the forces which act upon it are 
reducible to a single force which shall act in a direction perpen- 
dicular to the plane at the point of contact. 

2. When a power sustains a heavy body in equilibrio upon 
an inclined plane, then the power, the weight, and the pressure 
upon the plane, will be respectively, as the sine of the plane's 
inclination, the cosine of the angle which the direction of the 
power makes with the plane, and the cosine of the angle which 
the direction of the power makes with the horizon. 



STATICS : MECHANICAL POWERS. 219 

3. When the direction of the power is parallel to the plane, 
the power, weight, and pressure on the plane, are respectively 
as the height, length, and base of the plane ; or as the sine of 
inclination, radius, and cosine of inclination. 

Thus, suppose a plank 12 feet long were laid aslant from the 
ground to a window 4 feet high : then, since the length is three 
times the height, a power of 50 lbs. would sustain three times as 
much, or 150 lbs. upon the plank ; and a greater power, as ol 
55 or 60 lbs. would cause that weight to ascend. 

The truth of these propositions may be confirmed most 
readily, by attaching the weight to a chord tied to a spring steel- 
yard, by which the relations between the entire weight, and 
that supported by a chord either parallel to the inclined plane 
or in any other direction, may at once be measured. 

4. If two weights w, w f , sustain, each other upon two in- 
clined planes a c,cb, which have a common altitude c d, by means 
of a cord which runs freely over a pulley and is parallel to both 
planes, then will w:w'::ac:cb. 

5. When a heavy body is support- 
ed by two planes, as in the marginal 
figure, then, if the weight of the body 
be represented by the sine of the angle 
comprehended between the two planes, 
the pressures upon them are recipro- 
cally as the sines of the inclinations of 
those plane to the horizon : viz. 

The weight 
the pressure on a 
the pressure on b 

Thus, suppose the angle a b h was 30°, dbi 60°, and con- 
sequently a b d 90° : since the natural sines of 90°, 60°, and 
30°, are 1, '866, and \ respectively, or nearly as 100, 86*6, and 
50 ; if the heavy body weigh lOOlbs. the pressure upon a b 
would be 86 '6 lbs. and upon b d 50 lbs. 

This proposition is of very extensive utility, comprehending 
the pressure of arches on their piers, of buttresses against walls, 
or upon the ground, &c. because the circumstance of one of the 
planes becoming either horizontal, or vertical, will not affect 
the general relation above exhibited. 



Wedge, 
1. A wedge is a triangular prism, or a solid conceived to be 





220 



STATICS I MECHANICAL POWERS. 




generated by the motion of a plane triangle paral- 
lel to itself upon a straight line which passes 
through one of its angular points. The wedge is 
called isosceles, rectangular, or scalene, accord- 
ing as the generating triangle is isosceles, right- 
angled, or scalene. It is very frequently used in 
cleaving wood, as represented in the figure, and often in raising 
great weights. 

2. When a resisting body is sustained against the face of a 
wedge, by a force acting at right angles to its direction ; in the 
case of equilibrium, the power is to the resistance as the sine of 
the semi-angle of the wedge, to the sine of the angle which the 
direction of the resistance makes with the face of the wedge ; 
and the sustaining force will be as the cosine of the latter 
angle. 

3. When the resistance is made against the face of a wedge 
by a body which is not sustained, but will adhere to the place 
to which it is applied without sliding, the power is to the re- 
sistance, in the case of equilibrium, as the cosine of the differ- 
ence between the semi-angle of the wedge and the angle which 
the direction of the resistance makes with the face of the wedge, 
to radius. 

4. When the resisting body is neither sustained nor adheres 
to the point to which it is applied, but slides freely along 
the face of the wedge, the power is to the resistance as the 
product of the sines of the semi-angle of the wedge and the 
angle in which the resistance is inclined to its face to the square 
of radius. 



Screw. 




1. The screw is a spiral thread or groove cut 
round a cylinder, and every where making the 
same angle with the length of it. So that if the 
surface of the cylinder, with this spiral thread 
on it, were unfolded and stretched into a plane, 
the spiral thread would form a straight inclined 
plane, whose length would be to its height, as 
the circumference of the cylinder is to the distance between 
two threads of the screw : as is evident by considering that, in 
making one round, the spiral rises along the cylinder the dis- 
tance between the two threads. 

2. The energy of a power applied to turn a screw round, 
is to the force with which it presses upward or downward, 
setting aside the friction, as the distance between two threads 



STATICS .* MECHANICAL POWERS. 



221 




is to the circumference where the power is applied : viz. as 
circumf. of d c to dist. b i. 

3. The endless screw, or perpetual 
screw, is one which works in, and turns a 
dented wheel d f, without a concave or fe- 
male screw ; being so called because it may- 
be turned for ever, without coming to an 
end. From the diagram it is evident that 
while the screw turns once round, the 
wheel only advances the distance of one tooth. 

4. If the power applied to the lever, or handle of an endless 
screw, a b, be to the weight, in a ratio compounded of the pe- 
riphery of the axis of the wheel e h, to the periphery described 
by the power in turning the handle, and of the revolutions of 
the wheel d f to the revolutions of the screw c b, the power 
will balance the weight. Hence, 

5. As the motion of the wheel is very slow, a small power 
may raise a very great weight by means of an endless screw. 
And therefore the chief use of such a screw is, either where a 
great weight is to be raised through a little space ; or where 
only a slow gentle motion is wanted. For which reason it is 
very serviceable in clocks and watches. 

The screw is of admirable use in the mechanism of microme- 
ters, and in the adjustments of astronomical and other instru- 
ments of a refined construction. 

6. The mechanical advantage of a compound machine may be 
determined by analyzing its parts, finding the mechanical ad- 
vantage of each part severally, and then blending or compound- 
ing all the ratios. Thus, if m to 1, n to 1, r to 1, and s to 1, 
show the separate advantages ; then m n r s to 1, will measure 
the advantage of the system. 

7. The marginal representation of a common construction of a 
crane to raise heavy loads, will 
serve to illustrate this. By human 
energy at the handle a, the pinion 
b is turned ; that gives motion to 
the wheel w, round whose axle, c, 
a cord is coiled ; that cord passes 
over the fixed pulley, d, and thence 
over the fixed triple block, b, and 
the moveable triple block, p, be- 
low which the load, l, hangs. 
Now, if the radius of the handle 
be 6 times that of the pinion, the radius of the wheel w 10 times 
that of its axle, and a power equivalent to 30 lbs. be exerted 

30 u 2 




222 STATICS : MECHANICAL POWERS. 

at a ; then, since a triple moveable pulley gives a mechanical 
advantage of 6 to 1, we shall have 

30 X 6 X 10 X 6 = 10800 lbs. 
and such would be the load, l, that might be raised by a power 
of 30 lbs. applied at a, were it not for the loss occasioned by 
friction.* 



Section IV. — General application of the principles of Statics 
to the equilibrium of Structures. 

Every structure is exposed to the operation of a system of 
forces ; so that the examination of its stability involves the ap- 
plication of the general conditions of equilibrium. 

Now, no part of a structure can be dislocated, except it be 
either by a progressive, or a rotatory motion. For either this 
part is displaced, without changing its form, in which case it is 
as a system of invariable form, incapable of receiving any in- 
stantaneous motion, which is not either progressive or rotatory ; 
or else it happens to be displaced, changing at the same time its 
form : and this, considering the cohesion of tenacity, cannot 
take place, without the breaking of that part in its weakest sec- 
tion ; which generates a progressive motion, if the force acts 
perpendicularly to the section ; and a rotatory motion, if it acts 
obliquely. 

We shall here consider the most useful cases ; indicating by 
the word stress, that force which tends to give motion to the 
structure ; by resistance, that which tends to hinder it. 

Equilibrium of Piers. 

1. Taking the marginal figure for the 
vertical section of a pier, we may rea- 
son upon that section instead of the pier 
itself, if it be of uniform structure. 

Let g be the place of the centre of 
gravity, s r z the direction in which 
the stress acts, meeting x i, the ver- 
tical line through the centre of gra- 
vity, in i. Then, considering the stress as resolvable into 
two forces, one p, vertical, the other, q, horizontal ; the pier 
(regarding it as one body) can only give way either by a 

* We shall insert a selection of useful mechanical contrivances, after we have 
given the principles of dynamics. 




STATICS : EQUILIBRIUM OF PIERS. 223 

progressive motion from b towards a, or by a rotatory motion 
about a. 

2. The progressive motion is resisted by friction. If w de- 
note the weight of the pier, p the stress estimated vertically, 
and q its horizontal effort, then the pressure on the base 
= w 4- p, and its friction =f(w -f p), which is the amount of 
the resistance to progressive motion. So that to ensure sta- 
bility in this respect we must have 

/(W + P)>Q _. (1) 

While, to ensure stability in regard to rotation, we must have 

W.AX+P.AE> Q.ES (2) 

3. The second condition may be ascertained by a graphical 
process, thus : 

From the point a, let fall, on the direction of the stress, the 
perpendicular a z. Then, s being put for the whole stress, 
W . M X > s . A z. 

Or, suppose the two forces m and s to be applied at I, and 
complete the parallelogram, having sides which represent these 
forces. Then must the diagonal produced meet the base on the 
side of a, towards b, to ensure stability. 

4. If, as is very frequently the case, the vertical section of 
the pier is a rectangle, and s represent the specific gravity of 
the material of which it is constituted ; then the condition of 
the two kinds of equilibrium will be denoted by these two 
equations : viz. 

/. C B . A B . S = Q . . . . (3) . . . . A B 2 S = 2 Q . . (4) 

Example. Suppose a rectangular wall 39*4 feet high, and of 
a material whose specific gravity is 2000, is to sustain a hori- 
zontal strain of 9900 lbs. avoirdupois at its summit on the unit 
of length, 1 foot : what must be the thickness that there may be 
an equilibrium, taking/ = |. 

Here, that the wall may not be displaced horizontally, we 
must have 

a b > Q —/. s . c b > 9900 — § . -— - . 39*4 

16 

2000 X 39-4 3300 X 32 

64 < 39400 

52800 

> S9400 > 1-34 f6et 
And 2dly, that it may not be overturned, we must have 

ab> |2q > |2^ >v , 158 . 4>12 . 58feet . 

\ s \ 125 
Here, as the thickness required to prevent overturning is 
much the greatest, the computation in reference to the other 
kind of equilibrium may usually be avoided. 




224 statics : PRESSURE of earth against walls. 



Pressure of Earth against Walls, 

1. Let d a e f be the vertical sec- 
tion of a wall behind which is po- 
sited a bank or terrace of earth, of 
which a prism whose section is repre- 
sented by d a g would detach itself 
and fall down, were it not prevented 
by the wall. Then a g is denomi- 
nated the line of rupture or the natural slope, or natural de- 
clivity. In sandy or loose earth, the angle bag seldom 
exceeds 30° ; in stronger earth it becomes 37° : and in some 
favourable cases more than 45°. 

2. Now, the prism whose vertical section is d a g, has a 
tendency to descend along the inclined plane g a by reason of 
the force of gravity, g ; but it is retained in its place, 1st, by 
the force, q, opposed to it by the wall, and 2dly, by its cohesive 
attachment to the face A g, and by its friction upon the same 
surface. Each of those forces may be resolved into one, which 
is perpendicular to g a, and which is inoperative as to this in- 
quiry, and into another whose action is parallel to g a. The 
lines p i and i h, represent these composants of p, that force 
being represented by the vertical line p h, drawn from the 
centre of gravity p of the prism. The direction of the force q 
is represented by the horizontal line q h, and its composants 
by the lines q l, h l. The force that gives the triangle its ten- 
dency to descend is i h ; and the force opposed to this is l h 
together with the effects of cohesion and friction. Thus, 

i h = l h + cohesion + friction. 
It is evident, therefore, that the solution to this inquiry must 
be, in great measure, experimental. 

3. It has been found, however, theoretically, by M. Prony,* 
and confirmed experimentally, that the angle formed with the 
vertical by the prism of earth that exerts the greatest horizon- 
tal stress against a wall, is half the angle which the natural 
slope of the earth makes with the vertical : and this curious re- 
sult greatly simplifies the whole inquiry. 

The state of equilibrium is expressed by this equation : viz. 
|ad.ae 3 .s={ad 3 j. tan 3 % d a g. 
s and s representing the specific gravities of the wall and earth 
respectively. 

Example. The wall to be 39-37 feet high, of brick, specific 

* See a demonstration at p. 369, vol. ii. tenth edition of Dr. Hutton's Course of 
Mathematics. 



STATICS I PRESSURE OP EARTH AGAINST WALLS. 225 

gravity 2000, and the terrace of strong earth specific gravity 
1428, natural slope 53° from vertex. 
Then the above equation becomes 
h x 2 X 2000 X 39-37=4 X 39 ' 373 X 1428 X tan* 26h 

„;, | 1428 39-37 |1428 

or x = 39-37 tan 2Q°h - —-= — — 

\3X2000 2 \6000 

= 19-685 x -4878=9-6 feet, thickness of the wall. 

4. Of the experimental results the best which we have seen 
are those of M. Mayniel, from which the following are selected ; 
all along supposing the upper surface of the earth and of the 
wall which supports it, to be both in one horizontal plane. 

1st. Both theory and experiment indicate that the resultant 
h q of the thrust of a bank, behind a vertical wall, is at a dis- 
tance a q from the bottom of the wall=§ a d, the height. 

2dly. That the friction is half the pressure, in vegetable 
earths, four-tenths in sand. 

3dly. The cohesion which vegetable earths acquire, when 
cut in turfs, and well laid, course by course, diminishes their 
thrust by full two-thirds. 

4thly. The line of rupture behind a wall which supports a 
bank of vegetable earth is found at a distance d g from the in- 
terior face of the wall equal to *618 h, h being the height of the 
wall. 

5thly. When the bank is of sand, then d g=*677 h. 

6thly. When the bank is of vegetable earth mixed with small 
gravel, then d g=*646 h. 

7thly. If it be of rubbles, then d g=*414 h. • 

8thly. If it be of vegetable earth mixed with large gravel, 
then d G *618 h. 



Thickness of Walls, both faces vertical. 

1. Wall brick, weight of cubic foot =109 lbs. avoird. bank 
vegetable earth, carefully laid, course by course, d f=*16 h. 

2. Wall unhewn stones, 135lbs. per cubic foot, earth as be- 
fore, d f='15 h. 

3. Wall brick, earth clay well rammed, d f=*17 A. 

4. Wall unhewn stones, earth as above, d p=-16 h. 

5. Wall of hewn free stone, 170 lbs. to the cubic foot, bank 
vegetable earth, d p=-13 h ; if the bank be clay d f=*14 A. 

6. Bank of earth mixed with large gravel, 

Wall of bricks d f=*19 h. 

unhewn stone d p=*17 h. 

hewn free stone .... d p=*16 h. 



226 statics : PRESSURE op earth against walls. 

7. Bank of sand, 

Wall of bricks d p=-33 h. 

unhewn stone d p=*30 i 

hewn free stone .... d p=*26 h. 
When the earth of the bank or terrace is liable to be much 
saturated with water, the proportional thickness of wall must 
be at least doubled.* 

8. For walls with an interior slope, or a slope towards the 

bank, let the base of the slope be — of the height, and let s and 

n 

$, as before, be the specific gravities of the wall and of the 

earth ; then 

s h 

- + m ; 

7T s n 

where m = *0424, for vegetable or clayey earth mixed with 

large gravel ; m = -0464, if the earth be mixed with small 

gravel ; m = '1528, for sand ; and m = -166, for semi-fluid 

earths. ^ 

Example. — Suppose the height of a wall to be 20 feet, and 
2V of the height for the base of the talus or slope ; suppose, 
also, the specific gravities of the wall and of the bank to be 
2600, and 1400, and the earth semi-fluid : what, then, must be 
the thickness of the wall at the crown ? 

Here the theorem will become, 



d p == h f — 

\3 7 



= 20 f-L 

\1200 



DF = 20^ — - + *166. I|— *% 



= 20 v' -0008333 + '0894 — 1 = (20 X *3) — 1 
— 6 — 1=5 feet : while the thickness of the 
wall at bottom will be 6 feet. 



Equilibrium of Polygons. 

1. Let there be any number of lines, or bars, or beams, a b, 
bc, cd, de, &c. all in the same vertical plane, connected to- 
gether, and freely moveable about the joints or angles, a, b, c, d, 
e, &c. and kept in equilibrio by weights laid on the angles : It 
is required to assign the proportion of those weights : as also 
the force or push in the direction of the said lines ; and the ho- 
rizontal thrust at every angle. 

20 
* When weights of French cubic feet are given in kilogrammes, — of them will 

be the corresponding weight of an English cubic foot in pounds avoirdupois. 



STATICS .* PRESSURE OP EARTH AGAINST WALLS. 



227 




Through any point, 
as d, draw a vertical 
line a d h g, &c. ; to 
which, from any point, 
as c, draw lines in the 
direction of, or paral- 
lel to, the given lines 
or beams, viz. c a pa- 
rallel to a b, c b parallel to b c, c e to d e, c /to e p, c g to 
f g, &c. ; also c h parallel to the horizon, or perpendicular to 
the vertical line a d g, in which also all these parallels ter- 
minate. 

Then will all those lines be exactly proportional to the 
forces acting or exerted in the directions to which they are 
parallel, and of all the three kinds, viz. vertical, horizontal, 
and oblique. That is, the oblique forces or thrusts in direc- 
tion of the bars a b, b c, c d, d e, e f, p g, 

are proportional to their parallels c a, c b, c d, c e, cf, c g ; 
and the vertical weights on the angles b, c, d, e, f, &c. 
are as the parts of the vertical . . . . a b, b d, d e, ef,fg 9 
and the weight of the whole frame a b c d e f g, 
is proportional to the sum of all the verticals, or to a g ; also 
the horizontal thrust at every angle, is everywhere the same 
constant quantity, and is expressed by the constant horizontal 
line c h. 

Corol. 1. It is worthy of remark that the lengths of the 
bars a b, b c, &c. do not affect or alter the proportions of any 
of these loads or thrusts ; since all the lines c a, c b, a b, &c. 
remain the same, whatever be the lengths of a b, b c, &c* 
The positions of the bars, and the weights on the angles depend- 
ing mutually on each other, as well as the horizontal and oblique 
thrusts. Thus, if there be given the position of d c, and the 
weights or loads laid on the angles d, c, b ; set these on the ver- 
tical, d h, d by b a, then c b, c a, give the directions or posi- 
tions of c b, b a, as well as the quantity or proportion c h of the 
constant horizontal thrust. 

Corol. 2. If c h be made radius ; then it is evident that h a is 
the tangent, and c a the secant of the elevation of c a or a b 
above the horizon ; also h b is the tangent and c b the secant 
of the elevation of c b or c b ; also h d and c d the tangent 
and secant of the elevation of c d ; also h e and c e the tan- 
gent and secant of the elevation of c e or d e ; also h / and 
c / the tangent and secant of the elevation of e f ; and so on ; 
also the parts of the vertical a b, b d, ef, f g, denoting the 



228 EQUILIBRIUM OP ROOFS, &C. 

weights laid on the several angles, are the differences of the 
said tangents of elevations. Hence then in general, 

1st. The oblique thrusts, in the directions of the bars, are to 
one another, directly in proportion as the secants of their angles 
of elevation above the horizontal directions ; or, which is the 
same thing, reciprocally proportional to the cosines of the same 
elevations, or reciprocally proportional to the sines of the ver 
tical angles, a, b, d, e,f, g, &c. made by the vertical line with 
the several directions of the bars ; because the secants of any 
angles are always reciprocally in proportion to their cosines. 

2. The weight or load laid on each angle is directly propor- 
tional to the difference between the tangents of the elevations 
above the horizon, of the two lines which form the angle. 

3. The horizontal thrust at every angle is the same constant 
quantity, and has the same proportion to the weight on the top 
of the uppermost bar, as radius has to the tangent of the eleva- 
tion of that bar. Or, as the whole vertical a g, is to the line 
c h, so is the weight of the whole assemblage of bars, to the 
horizontal thrust. 

4. It may hence be deduced also, that the weight or pressure 
laid on any angle, is directly proportional to the continual pro- 
duct of the sine of that angle and of the secants of the elevations 
of the bars or lines which form it. 

Scholium. This proposition is very fruitful in its practical 
consequences, and contains the whole theory of centerings, and 
indeed of arches, which may be deduced from the premises by 
supposing the constituting bars to become very short, like arch 
stones, so as to form the curve of an arch. It appears too, that 
the horizontal thrust, which is constant or uniformly the same 
throughout, is a proper measuring unit, by means of which to 
estimate the other thrusts and pressures, as they are all deter- 
minable from it and the given positions ; and the value of it, as 
appears above, may be easily computed from the uppermost or 
vertical part alone, or from the whole assemblage together, or 
from any part of the whole, counted from the top down- 
wards. 

In all the useful cases, a model of the structure may be 
made, and the relations of the pressures at any angle, whether 
horizontal, vertical, or in the directions of the beams, may be 
determined by a spring steel-yard applied successively in the 
several directions, after the manner described in Art. 4. Sect. 1. 
Statics. 

2. If the whole figure in the preceding problem be inverted, 
or turned round the horizontal line a g as an axis, till it be 



EQUILIBRIUM OF ROOFS, &C. 



229 



completely reversed, or in the same vertical plane below the 
first position, each angle d, d, &c. being in the same plumb 
line ; and if weights i, k, I, m, n, which are respectively equal 
to the weights laid on the angles b, c, d, e, f, of the first 
figure, be now suspended by threads from the corresponding 
angles h, c, d, e, f, of the lower figure ; those weights keep 
this figure in exact equilibrio, the same as the former, and all 
the tensions or forces in the latter case, whether vertical or ho- 
rizontal or oblique, will be exactly equal to the corresponding 
forces of weight or pressure or thrust in the like directions of 
the first figure. 




This, again, is a proposition most fertile in its conse- 
quences, especially to the practical mechanic, saving the 
labour of tedious calculations, but making the results of ex- 
periment equally accurate. It may thus be applied to the prac- 
tical determination of arches for bridges, with a proposed 
road way ; and to that of the position of the rafters in a curb or 
mansard roof. 

3. Thus, suppose it were required to make such a roof, with 
a given width a e, and of four proposed rafters ab,bc,cd,de. 
Here, take four pieces that are equal or 
in the same given proportions as those 
proposed, and connect them closely to- 
gether at the joints a, b, c, d, e, by pins 
or strings, so as to be freely moveable 
about them ; then suspend this from two 
pins, a e, fixed in a horizontal line, and 
the chain of the piece will arrange itself 
in such a festoon or form, a b c d e, that 
all its parts will come to rest in equili- 
brio. Then, by inverting the figure, it will exhibit the form and 
31 X 




230 



STATICS : MANSARD ROOF. 



frame of a curb roof a b c d e, which will also be in equilibrio, 
the thrusts of the pieces now balancing each other, in the same 
manner as was done by the mutual pulls or tensions of the 
hanging festoon a b c d e. 

4. If the mansard be constituted of four equal rafters ; then, 
if angle c A e = m, angle c a b = x ; it is demonstrable that 
2 sin 2 x = sin 2 m. So that if the span a e, and height m c, 
be given, it will be easy to compute the lengths a b, b c, &c. 

Example. Suppose a e = 24 feet, m c 12. 
m c 
Then rr~~ = 1 = tan 45° angle c a m = m. 

,\ sin 2 m = sin 90° = 1, and sin 2 x = h 
.-. 2 x = 30°, and x = 15° = c a b 
Hence m a b = 45° + 15° = 60° 

andMBA = § (180° — 2 x 15°) =90° — 15° = 75° 
and a M b= 180° — (75° +60°) = 45° 
Lastly, sin 75° : sin 45 :: a m == 12 : a b = 8*7846 feet. 

Note. In this example, since A m = m c, as well as a b = b c, 
it is evident that m b bisects the right angle a m c ; yet it 
seemed preferable to trace the steps of a general solution. 

Stability of Arches. 

1. If the effect of the force of gravity upon the ponderating 
matter of an arch and pier, be considered apart from the ope- 
ration of the cements which unite the stones, &c. the investi- 
gation is difficult to practical men, and it furnishes results 
that require much skill and care in their application. But, 
in an arch whose component parts are united with a very 
powerful cement, those parts do not give way in vertical 
columns, but by the separation of the entire mass, including 
arches and piers, into three, or, at most, into four parts. In 
this case, too, the conditions of equilibrium are easily ex- 
pressed and easily applied. 

Let/ f, /' f', be the 
joints of rupture, or places 
at which the arch would 
most naturally separate, 
whether it yield in two 
pieces or in one. Let g 
be the centre of gravity of 
the semi-arch /f k k, and 
g' that of the pier a b f/ 
Let f i be drawn parallel 
to the horizon, and g h be 
demitted perpendicularly 




statics: arches and piers. 231 

upon it ; also let g' d be a perpendicular passing through g', 
and f e drawn from f parallel to it. Then 

2. Prop. If the arch f? f' /' tend to fall vertically in one 
piece, removing the sections/ F,/' f' ; if a be the weight of 
the semi-arch /f x &, and p that of the pier up to the joint 
f f, the equilibrium will be determined by these two equa- 
tions : viz. 



/•»-* e-r/) (1) 

AD /CI A E\ /oX 

P. =A( ) (2) 

F E \F I F E/ 



where f is the measure of the friction, or the tangent of the 
angle of repose of the material, and the first equation is that of 
the equilibrium of the horizontal thrusts, while the latter indi* 
cates the equilibrium of rotation about the exterior angle a of 
the pier. 

3. Prop. If each of the two semi-arches f k, k, f', tend to 
turn about the vertex k of the arch, removing the points f, f', 
the equilibrium of horizontal translation, and of rotation, will 
be respectively determined by the following equations : viz. 

/•*-*(£?-■')' < 3 > 

AD /F H AE\ , . y 

P. =A(- ) .... (4) 

F E > K I F E / 

4. Hence it will be easy to examine the stability of any ce- 
mented arch, upon the hypothesis of these two propositions. 
Assume different points, such as f in the arch, for which let the 
numerical values of the equations (1) and (2), or (3) and (4), be 
computed. To ensure stability, the first members of the re- 
spective equations must exceed the second ; those parts will be 
weakest, where the excess is least. 

If the figure be drawn on a smooth drawing pasteboard, upon 
a good sized scale, the places of the centres of gravity may be 
found experimentally, as well as the relative weights of the 
semi-arch and piers, and the measures of the several lines from 
the scale employed in the construction. 

If the dimensions of the arch were given, and the thickness 
of the pier required, the same equations would serve ; and dif- 
ferent thicknesses of the pier might be assumed until the first 
members of the equations come out largest. 



232 statics : models. 

The same rules are applicable to domes, simply taking the 
imgulas instead of the profiles. 



Models. 

From an experiment made to ascertain the firmness of the 
model of a machine, or of an edifice, certain precautions are 
necessary before we can infer the firmness of the structure 
itself. 

The classes of forces must be distinguished ; as whether they 
tend to draw asunder the parts, to break them transversely, or 
crush them by compression. To the first class belongs the 
stretching suffered by key-stones, or bonds of vaults, &c. : to 
the second, the load which tends to bend or break horizontal or 
inclined beams ; to the third the weight which presses verti- 
cally upon walls and columns. 

Prob. I. If the side of a model be to the corresponding side 
of the structure, as 1 to n, the stress which tends to draw asun- 
der, or to break transversely the parts, increases from the 
smaller to the greater scale, as 1 to n 3 ; while the resistance of 
those ruptures increases only as 1 to n 2 . 

The structure, therefore, will have so much less firmness than 
the model as n is greater. 

If w be the greatest weight which one of the beams of the 

model can bear, and w the weight or stress which it actually 

w 
sustains, then the limit of n will be n — — . 

w 

Prob. 2. The side of the model being to the corresponding 
side of the structure as 1 to n, the stress which tends to crush 
the parts by compression, increases from the smaller to the 
greater scale, as 1 to n 3 , while the resistance increases only in 
the ratio of 1 tow. 

Hence, if w were the greatest load which a modular wall or 

column could carry, and w the weight with which it is actually 

loaded ; then the greatest limit of increased dimensions would 

. w 

be found from the expression n = </— , 

If, retaining the length or height n h, and the breadth n b, 

we wished to give to the solid such a thickness x t, as that it 

should not break in consequence of its increased dimensions, 

w 
we should have x = n 2 y/— . 

w 

In the case of a pilaster with a square base, or of a cylin- 



STATICS : MODELS. 



233 



drical column, if the dimension of the model were d 9 and of 
the largest pillar, which should not crash with its own weight 
when n times as high, x d, we should have 



3 \n 2 w 
\ w 



These theorems will often find their application in the profes- 
sion of an architect or an engineer, whether civil or military. 

3. Suppose, for an example, it were required to ascertain the 
strength of Mr. Smart's "Patent Mathematical Chain-bridge," 
from experiments made with a model. In this ingenious con- 
struction, the truss-work is carried across from pier to pier, 
so that the road-way from a to b, and thence entirely across, 
shall be in a horizontal plane, and all the base bars, diagonal 
bars, hanging bars, and connecting bolts, shall retain their own 
respective magnitudes throughout the structure. The an- 
nexed representation of half the bridge so exhibits the con- 
struction as to supersede the necessity of a minute verbal de- 
scription. 




Now, let / represent the horizontal length of the model, 
(say 12 feet,) from interior to exterior of the two piers, w its 
weight (say 30 pounds), w the weight it will just sustain at 
its middle point b before it breaks (say 350 lbs.) Let n I the 
length of a bridge actually constructed of the same material as 
the model, and all its dimensions similar : then, its weight will 
be n 3 w, and its resisting power to that of the model, as n* to 1, 
being = n 2 (w + h w.) Hence n 2 (w + h w) — %n s w = n* 
w — i n 2 (n— 1) w, the load which the bridge itself would 
bear at the middle point. 

x2 



234 statics : MODELS. 

Suppose n = 20, or the bridge 240 feet long, and entirely 
similar to the model ; then we shall have (400 x 350) — 200 
(20 — 1) 30 = 140000 — 114000 == 26000 lbs. = 11 tons \2\ 
cwt., the load it would just sustain in the middle point of its 
extent. 

Note. — This bridge is, in fact, a suspension bridge, and would 
require brace or tie-chains at each pier. A considerable im- 
provement upon its construction, by Colonel S. H. Long, of the 
American Engineers, is described in the Mechanic's Magazine? 
vol. xiii. or No. 368. 



DYNAMICS. 235 



CHAPTER X. 
DYNAMICS. 



1. The mass of a body is the quantity of matter of which it 
is composed. 

The knowledge of the mass of a body is given to us by that 
property of matter which we call inertia ; and which being 
greater or less as the mass is greater or less, we regard as an in- 
dex of the mass itself. 

2. Density is a word by which we indicate the comparative 
closeness or otherwise of the particles of bodies. Those bodies 
which have the greatest number of particles, or the greatest 
quantity of matter, in a given magnitude, we call most dense ; 
those which have the least quantity of matter, least dense ; 
Density and weight are regarded as correlatives ; so that the 
heaviest bodies of a given size, are the most dense, the lightest 
bodies, the least dense. 

Thus lead is more dense than freestone ; freestone more 
dense than oak ; oak more dense than cork. 

3. When bodies are impelled by certain forces, they receive 
certain velocities, and move over certain spaces, in certain 
times. So that body, force, velocity, space, time, are the sub- 
jects of investigation in Dynamics ; and in mathematical theo- 
rems, they are usually represented by the initial letters, b,f, v, 
s, t : or, if two or more bodies, &c. are compared, two or more 
corresponding letters b, b, b', v, v, v', &c. are employed in the 
formulae. Gravity, which is a separate force incessantly acting, 
is represented by g ; and momentum, or quantity of motion, 
by m, this being the effect produced by a body in motion. 

Force is distinguished into motive and accelerative, or 
retardive. 

4. Motive force, otherwise called momentum, or force of 
percussion, is the absolute force of a body in motion, &c. ; and 
is expressed by the product of the weight or mass of matter in 
the body multiplied by the velocity with which it moves. But 

5. Accelerative force, or retardive force, is that which 
respects the velocity of the motion only, accelerating or 
retarding it ; and it is denoted by the quotient of the motive 



236 DYNAMICS : UNIFORM MOTIONS. 

force divided by the mass or weight of the body. So, if a body 
of 2 lbs. weight be acted upon by a motive force of 40, the ac- 
celerating force is 20 ; but, if the same force of 40 act upon 
another body of 4 lbs. weight, the accelerating force is then only 
10 ; that is, it is only half the former, and will produce only 
half the velocity. 



Section I. — Uniform Motions, 

1. The space described by a body moving uniformly, is re- 
presented by the product of the velocity into the time : and in 
comparing two, we say 

s : s :: t v : t v. 

2. In regard to momenta, m varies, as b v 9 or 

m : m : : b v : b v. 

Example. Two bodies, one of 10, the other of 5 pounds, 
are acted upon by the same momentum, or receive the same 
quantity of motion 30. They move uniformly, the first for 8 
seconds, the second for 6 ; required the spaces described by 
both. 

Here 4£ = 3 = v, and 3 T ° = 6 = v 

Then t v = 3 x 8 = 24 = s ; and t v = 6 X 6 = 36 =5. 
Thus the spaces are 24 and 36 respectively. 



Section II. — Motion uniformly accelerated. 

1. Motion uniformly accelerated, is that of a material point 
or body subjected to the continual action of a constant force. 

2. In this motion the velocity acquired at the end of any 
time whatever is equal to the product of the accelerating 
force into the time ; and the space described is equal to the 
product of half the accelerating force into the square of the 
time. 

3. The spaces described in successive seconds of time are as 
the odd numbers, 1, 3, 5, 7, 9, &c. 

4. Gravity is a constant force, whose effect upon a body fall- 
ing freely in a vertical line is represented by g ; and the motion 
of such body is uniformly accelerated. 



DYNAMICS *. MOTION UNIFORMLY ACCELERATED. 237 

5. The following theorems are applicable to all cases of mo- 
tion uniformly accelerated by any constant force /. 



v = -^- = gft = s/2 gfs 

t __ 2 s _ v I s _ 

~v gf~S i gf* 
n v 2 s v 2 



gt g t* 2gs 

Hence, in all motions of this nature, as soon as the ratio of 
the force, f, to the force of gravity, g, is known, the circum- 
stances of space, time, velocity, &c. may be computed ; or con- 
versely, knowing the space described in a given time, or the 
velocity acquired at the end of such time, the value of f may 
be obtained. 

6. When the force of gravity acts freely, as when a body 

falls in a vertical line, f is omitted in the theorems, and we 

have 

v 2 
s = hgt*=—=h tv 
2g 



V — 


-gt 


2 s 
~ t 


= s/2 


t= 


V 


2 s _ 


>/-* 




g 


V 


g 


g-- 


V 

" T 


_ 2 s _ 

~~ T~~~ 


~ 2 s 



7. Now, it has been ascertained by very accurate experi- 
ments that a body in the latitude of London falls nearly 16 T ^ 
feet in the first second of time, and that at the end of that time 
it has acquired a velocity double, or of 32£ feet ; therefore, if 
i g denote 16^ feet, the space fallen through in one second 
of time, or g the velocity generated in that time ; then, if the 
first series of natural numbers be seconds of time, 

namely, the times in seconds 1", 2", 3", 4", &c. 

the velocities in feet will be 321, 64§, 96i, 128|, &c. 

the spaces in the whole times 16^, 64§, 144|, 257s, &c. 

and the space for each second 16 t l, 48?, 80 T 5 2, 112 r 7 -,&c. 
32 



238 DYNAMICS : DESCENTS BY GRAVITY. 

of which spaces the common difference is 32 a feet, the natural 
and obvious measure of g, the force of gravity. 

8. If, instead of a heavy body being allowed to fall freely, 
it be propelled vertically upwards or downwards with a given 
velocity, v, then 

s = t v =F h g t 2 ; 
an expression in which the upper sign — must be taken when 
the projection is upwards, the lower sign + when the projec- 
tion is downwards. 

When only an approximate result is required with reference 
to bodies falling vertically, 32 may be put for g, instead of 32|: 
there would then result, in motions from quiescence, 

v 2 
s = 16 t*= — = % tv 
64 

, v . 2s 

2 s 
v = 8 y/s = — - = 32 t. 

Thus, if the space descended were 64 feet, we should have 
v = 8 X 8 = 64, and t = § = 2 seconds. 

If the space descended were 400 : then v = 8 x 20 == 160, 
and t == y = 5. 

9. The force of gravity differs a little at different latitudes ; 
the law of the variation is not as yet precisely ascertained ; but 
the following theorems are known to represent it very nearly. 
That is, if g denote the force of gravity at latitude 45°, g' the 
force at any other place : then 

g'=g (1* — -002837 cos 2 lat.) 
g'=g (1 + -002837), at the poles. 
g'=g (1 — -002837), at the equator. 

10. Motion over a fixed pulley. — In this case let the two 
weights which are connected by the cord that goes over the 

•yy W 

pulley be denoted by w and w : then = / in the for- 

w + w 



mulae of art. 5 ; so that 

w — w 



w + w 



igf- 



Or, if the resistance of the friction and inertia of the pulley 
be represented by r ; then 

s = . \ g p. 

w -f- w + r 



DYNAMICS : MOTION OVER PULLEYS, &C. 



239 



Example. — Suppose the two weights to be 5 and 3 lbs. re- 
spectively, what will be the space descended in 4 seconds ? 

16 = i . 16 T ' T 



Here 



«\ XW3- 5 - 3 

2 gt — 



w + w 
16^ . 4 = 64h feet. 



5+3 



16 T V 



16 = 



Example II. — But, suppose that in an actual experiment with 
two weights of 5 and 3 lbs. over a pulley, the heavier weight 
descended only 50 feet in 4 seconds. 



Then 



w 



to 



h g t* = 50 feet : and, as w, w, g, and /, 



w + w + r 
are the same in both examples, 

we have w+w + r : w + w :: 64| : 50 
or, dividendo r : w-\-w :: 14| : 50 
that is, r: 5 + 3 :: 14£ : 50 

whence r = ^±^1^= iM = 2-2933 lbs. 
50 50 

the measure of the resistance and the inertia. 

Note. — Similar principles are applicable in a variety of other 
cases : and by varying the relations of w, w, and r, the force 
may have any assigned ratio to that of gravity ; which is, in- 
deed, the foundation of Mr. Atwootfs elegant apparatus for ex- 
periments on accelerating forces. The inquisitive reader may 
see an account of it in the 2d vol. of my Mechanics ■, or in al- 
most any of the general dictionaries of arts and sciences. 



11. If, instead of pulleys, small 
wheels and axles, as in the margi- 
nal figure, be employed, to raise 
weights by the preponderance of 
equal weights: then, if the dia- 
meter of wheel and axle a be as 3 
to 2 ; those of wheel and axle b, as 
5 to 2 ; those of c, as 8 to 2 ; it 
will be found that the weight b will 
be elevated more rapidly than 
either a or c : the proportion of 5 
to 2, (or, correctly, of 1 + ^/2 to 1) 
being in that respect the most fa- 
vourable. 




240 dynamics: motion on inclined planes. 



Motion on inclined Planes. 

1. When bodies move down inclined planes, the accelerating 
force (independently of the modification occasioned by the po- 
sition of the centre of gyration) is expressed by—, the quo- 

V 

tient of the height of the plane divided by its length, or by 
what is equivalent, the sine of the inclination of the plane, 
that is to say, sin i. In this case, therefore, the formulae be- 
come 

v 2 

1 . s = h g t 2 sin i = : — : = h t v 

2 g sin % 

% s 

2 . v = g t sin i = -s/(2 g s sin i) = — 



\g8ini v 



Farther, if v be the velocity with which a body is projected 
up or down a plane, then 

4 . v = v =F g t sin i 

v 2 — v* 

5 . s = V t =F 2 g t sin l = ;. 

2 g sin i 

Making v = 0, in equa. 4, and the latter member of equation 
5, the first will give the time at which the body will cease to 
rise, the latter the space. 

Example. — Suppose a body be projected up a smooth in- 
clined plane whose height is 12 and length 193 feet, with a ve- 
locity of 20 feet per second, how high will it rise up the plane 
before its motion is extinguished ? 

V 2_ v * 400 — 400 

Here s = : — - becomes s = 



2 g sin i 64£. T W JL93 J± _ 

3 ' 19S 

400 /.i . , 

=s =100 feet, the space required. 

2. With regard to the velocities acquired by bodies in fall- 
ing down planes of the same height, this proposition holds ; 
viz. that they are all equal, estimated in their respective di- 
rections. Thus, if 
a d, b e, c f, be 
grooves of different 
inclinations, and a c, 
d f, horizontal lines, 
the balls a, b, c, after 




DYNAMICS : MOTION ON INCLINED PLANES. 



241 




descending through those planes will have equal velocities 
when they arrive at d, e, f, respectively. 

3. Also, all the chords, such as a d, 
b d, c d, that terminate either in the 
upper or the lower extremity of the 
vertical diameter of a circle, will be 
described in the same time by heavy 
bodies a, b, c, running down them ; 
and that time will be equal to the time 
of vertical descent through the diame- 
ter. 

4. If three 
weights, as a, b, 
c, be drawn up 
three planes of 
different inclina- 
tions, by three 
equal weights 
hanging from 
cords over pulleys at p, then if the length of the middle plane 
be twice its height, the body b will be drawn up that plane, 
quicker than either of the other weights A or c. 

Or, generally, to ensure an ascent up a plane in the least time, 
the length of the plane must be to its height, as twice the 
weight to the power employed. 

5. If it be proposed to construct a roof over a building of a 
given width, so that the rain shall run quickest off it, then each 
side of the roof must be inclined 45° to the horizon, or the an- 
gle at the ridge must be a right angle. 

6. The force by which spheres, cylinders, &c. are caused to 
revolve as they move down an inclined plane (instead of sliding) 
is the adhesion of their surfaces occasioned by the pressure 
against the plane : this pressure is part of the body's weight ; 
for the weight being resolved into its components, one in the 
direction of the plane, the other perpendicular to it, the latter 
is the force of the pressure ; and, while the same body rolls 
down the plane, will be expressed by the cosine of the plane's 
elevation. Hence, since the cosine decreases while the arc or 
angle increases, after the angle of elevation arrives at a certain 
magnitude, the adhesion may become less than what is neces- 
sary to make the circumference of the body revolve fast enough ; 
in this case the body descends partly by sliding and partly by 
rolling. And the same may happen in smaller elevations, if 
the body and plane are very smooth. But at all elevations the 

Y 



242 DYNAMICS : MOTION ON INCLINED PLANES. 

body may be made to roll by the uncoiling of a thread or rib 
band wound about it. 

If w denote the weight of a body, s the space described by 
a body falling freely, or sliding freely down an inclined plane, 
then the spaces described by rotation in the same time by the 
following bodies, will be in these proportions. 

1. A hollow cylinder, or cylindrical surface, s = h s tension 
of the cord in the first case = § w. 

2. In a solid cylinder, s = •§ s, tens. ={w. 

3. In a spheric surface, or thin spherical shell, s = | s, 
tens. =|w. 

4. In a solid sphere, s = \ s, tens. = f w. 

If two cylinders be taken of equal size and weight, and with 
equal protuberances upon which to roll, as in the marginal 
figures : then, if lead be coiled uniformly over 
the curve surface of b, and an equal quantity 
of lead be placed uniformly from one end to 
the other near the axis in the cylinder a, that 
cylinder will roll down any inclined plane 
quicker than the other cylinder b. The rea- 
son is that each particle of matter in a roll- 
ing body, resists motion in proportion to 
the square of its distance from the axis of 
motion : and the particles of lead which most 
resist motion are placed at a greater distance from the axis in 
the cylinder b than in a. 

7. The friction between the surface of any body and a 
plane, may be easily ascertained by gradually elevating the 
plane until the body upon it just begins to slide. The friction 
of the body is to its weight as the height of the plane to its 
base, or as the tangent of the inclination of the plane to the 
radius. Thus, if a piece of stone in weight 8 pounds, just 
begins to slide when the height of the plane is 2 feet, and its 
base 2\ : then the friction will be -f- the weight, or | of 8 lbs. 
= 6 $ lbs. 

8. After motion has commenced upon an inclined plane, the 
friction is usually much diminished. It may easily be ascer- 
tained experimentally, by comparing the time occupied by a 
body in sliding down a plane of given height and length, or 
given inclination, with that which the simple theorem for /, 
would give. For, if/ be the value of the friction in terms of 
the pressure, the theorem for t will be 

J 2 s \ 2 s 

—r-- — : — 77-, instead of £ == I — : — :. Hence 
g(smi— fY \#sinz 

t' 2 : t* :: sin i : sin i—f 




DYNAMICS : MOTION ON INCLINED PLANES. 



243 



Example. — Suppose that a body slides down a plane in 
length 30 feet, height 10, in 2| seconds, what is the value ot 
the friction. 

| 2 s I 60 

Here t — \ — : — r — k I nn , _ x = 2*366 nearly. 



32i 



^ g sin i 
Hence (2'6)» : (2-366) 2 :: I : '27603 = sin i -f 

Consequently, -33333 — -27603 = -0573 value of the fric- 
tion, the weight being unity. 

9. When a weight is to be moved 
either up an inclined plane, or along 
an horizontal plane, the angle of 
traction p w b, that the weight may 
be drawn with least effort, will vary 
with the value of f The magnitude 
of that angle p w b for several values 
of/ are exhibited below. 




/ 


P W B 


/ 


P W B 


/ 


P W B 


/ 


P W B 


/ 


P W B 


i 


P W B 


1 


45° 0' 


i 

2 


26°34' 


i 


18°26' 


i 

4 


14° 2' 


i 

5 


11°19' 


9°28' 


4 
5 


38 40 


4 


23 58 


_4_ 


16 54 


4 


13 15 


4 


10 47 


a 

7 


8 8 


2 


33 41 


2 


21 48 


2 
T 


15 57 


2 


12 32 


3 
1 1 


10 18 


1 
B 


7 8 


f 


29 45 


A 


19 59 


4 
15 


14 56 


4 


11 53 


4 

23 


9 52 


9" 


6 20 



10. If, instead of seeking the line of traction so that the 
moving force should be a minimum, we required the position 
such that the suspending force to keep a load from descending 
should be a minimum, or a given force should oppose motion 
with the greatest energy ; then the angles in the preceding 
table will be still applicable, only the angle in any assigned 
case must be taken below, as b w p. This will serve in the 
building and fastening walls, banks of earth, fortifications, &c. 
and in arranging the position of land-lies, &c. 



Section III. — Motions about a Centre or Jlxis. 

Pendulum, simple and compound; Centres of Oscillation, 
Percussion, and Gyration. 

Def. 1. The centre of oscillation is that point in the axis of 
suspension of a vibrating body in which, if all the matter of 
the system were collected, any force applied there would ge- 
nerate the same angular velocity in a given time as the same 



244 DYNAMICS : LINE OF FRACTION. 

force at the centre of gravity, the parts of the system revolving 
in their respective places. 

Or, since the force of gravity upon the whole body may be 
considered as a single force (equivalent to the weight of the 
body) applied at its centre of gravity, the centre of oscillation 
is that point in a vibrating body into which, if the whole were 
concentrated and attached to the same axis of motion, it would 
then vibrate in the same time the body does in its natural state. 

Cor. From the first definition it follows that the centre ot 
oscillation is situated in a right line passing through the cen- 
tre of gravity, and perpendicular to the axis of motion. It is 
always farther from the point of suspension than the centre of 
gravity. 

Def. 2. The centre of gyration is that point in which, if all 
the matter contained in a revolving system were collected, the 
same angular velocity will be generated in the same time by a 
given force acting at any place as would be generated by the 
same force acting similarly in the body or system itself. 

When the axis of motion passes through the centre of gravity, 
then is the centre called the principal centre of gyration. 

The distance of the centre of gyration from the point of sus- 
pension, or the axis of motion, is a mean proportional between 
the distances of the centres of oscillation and gravity from the 
same point or axis. 

If s represent the point of suspension, g the place of the cen- 
tre of gravity, o that of the centre of oscillation, and r that of 
the centre of gyration. Then 

sr = ^/so.sg (1) 

and s o . s g = a constant quantity for the same body and the 
same plane of vibration. 

Def. 3. The Centre of Percussion is that point in a body 
revolving about an axis, at which, if it struck an immoveable 
obstacle, all its motion would be destroyed, or it would not in- 
cline either way. 

When an oscillating body vibrates with a given angular velo- 
city, and strikes an obstacle, the effect of the impact will be the 
greatest if it be made at the centre of percussion. 

For, in this case the obstacle receives the whole revolving 
motion of the body ; whereas, if the blow be struck in any other 
point, a part of the motion of the pendulum will be employed 
in endeavouring to continue the rotation. 

If a body revolving on an axis strike an immovable obstacle 
at the centre of percussion, the point of suspension will not be 
affected by the stroke. 

We can ascertain this property of the point o when we 
give a smart blow with a stick. If we give it a motion 



DYNAMICS : PENDULUMS, &C. 245 

round the joint of the wrist only, and, holding it at one extre- 
mity, strike smartly with a point considerably nearer or more 
remote than § of its length, we feel a painful wrench in the 
hand : but if we strike with that point which is precisely at § 
of the length, no such disagreeable strain will be felt. If we 
strike the blow with one end of the stick, we must make its 
centre of motion at § of its length from the other end ; and then 
the wrench will be avoided. 

Prop. The distance of the centre of percussion from the axis 
of motion is equal to the distance of the centre of oscillation 
from the same : supposing that the centre of percussion is re- 
quired in a plane passing through the axis of motion and cen- 
tre of gravity. 

Def. 4. A Simple Pendulum, theoretically considered, is a 
single weight, regarded as a point, or as a very small globe, 
hanging at the lower extremity of an inflexible right line, void 
of weight, and suspended from a fixed point or centre, about 
which it oscillates. 

Def. 5. A Compound Pendulum is one that consists of se- 
veral weights moveable about one common centre of motion, 
but so connected together as to retain the same distance 
both from one another and from the centre about which they 
vibrate. 

Or any body, as a cone, a cylinder, or of any shape, regular 
or irregular, so suspended as to be capable of vibrating, may be 
regarded as a compound pendulum ; and the distance of its cen- 
tre of oscillation from any assumed point of suspension, is con- 
sidered as the length of an equivalent simple pendulum. 

Any such vibrating body will have as many centres of oscil- 
lation as you give it points of suspension ; but when any one 
of those centres of oscillation is determined, either by theory 
or experiment, the rest are easily found by means of the pro- 
perty that s o . s g is a constant product, or of the same value 
for the same body. 

Def. 6. When a body either revolves about an axis, or oscil- 
lates, the sum of the products of each of the material elements, 
or particles of that body, into the squares of their respective 
distances from the axis of rotation, is called the momentum of 
inertia of that body. (See art. 6, p. 241). 

A point, or very small body, on descending along the suc- 
cessive sides of a polygon in a vertical plane, loses at each angle 
a part of its actual velocity equal to the product of that velocity 
into the versed sine of the angle made by the side which it has 
just quitted, and the prolongation of the side upon which it is 
just entering. 

33 Y 2 



246 



DYNAMICS : PENDULUMS, &C. 



Therefore that loss is indefinitely small in curves. 

7. A heavy body which descends along a curve posited in 
a vertical plane, by the force of gravity, has, in any point what- 
ever, the same velocity as it would have if it had fallen through 
a vertical line equal to that between 
the top and the bottom of the arc run 
over : and when it has arrived at the 
bottom of any such curve, if there 
be another branch either similar or 

dissimilar, rising on the opposite side, the body will rise along 
that branch (apart from the consideration of friction) until it 
has reached the horizontal plane from which it set out. Thus, 
after having descended from a to v, it will have the same velo- 
city as that acquired by falling through d v, and it will ascend 
up the opposite branch until it arrives at b. 

8. If the body describe a curve by a pendulous motion, the 
same property will be shown, free from the effects of friction. 
Thus, let a ball hang 
by a flexible cord s d 
from a pin at s : then, 
after it has descended 
through the arc d e, 
it will pass through 
an equal and similar 
arc e a, going up to a 
in the same horizontal 
line with d, and as- 
cending from e to a in an interval of time equal to that which 
it descended from d to e. But, if a pin projecting from p or p 
stop the cord in its course, the ball will still rise to b or to c, in 
the same horizontal line with a and d ; but will describe the 
ascending portions of the curve in shorter intervals of time than 
the descending branch. 

9. When a pendulum is drawn from its vertical position, it 
will be accelerated in the direction of the tan- 
gent of the curve it would describe, by a force 
which is as the sine of its angular distance 
from the vertical position. Thus, the accele- 
rating force at a, would be to the accelerating 
force at b, as a f to b e. (See art. 5, on the 
Centre of Gravity). 

This admits of an easy experimental proof. 

10. If the same pendulous body descend through different 
arcs, its velocity at the lowest point will be proportional to the 
chords of the whole arcs described. Thus, the velocity at d, 




/ 



DYNAMICS : PENDULUMS, &C. 247 

after passing through a b d, will be to the velocity at d after 
descending through the portion b d only, as a d to b d. 

11. Farther, velocity after describing a b d, is to velocity 
after describing b d, as \/f d to ^/e d. 

If, therefore, we would impart to a body a given velo- 
city v, we have only to compute the height f d, such that 

v 2 v 2 

f d = — = — r-^ — , and through the point f draw the hori- 
2 g 64§ feet & r 

zontal line f a ; then, letting the body descend as a pendulum 

through the arch a b d, when arrived at d, it will have acquired 

the proposed velocity. 

This is extremely useful in experiments on the collision of 

bodies. 

12. The oscillations of pendulums in any arcs of a cycloid 
are isochronal, or performed in equal times. 

13. Oscillations in small portions of a circular arc are iso- 
chronal. 

14. The numbers of oscillations of two different pendulums, 
in the same time, and at the same place, are in the inverse ratio 
of the square roots of the length of these pendulums. 

15. If / be the length of a single pendulum, or the dis- 
tance from the point of suspension to the centre of oscil- 
lation in a compound pendulum, g == the measure of the 
force of gravity (32^ feet, or 386 inches at the level of St. 
Paul's* in the latitude of London), t the time of one oscil- 
lation in an indefinitely small circular arc, and * = 3*141593 : 
then 



■-£ 



pendulum 
in lat. 

of 
London. 



16. Conformably with this we have 
39s inches, length of the second 

9|| inches, half second 

4^| inches, third of second 

2 T 5 ¥ 7 ¥ inches, quarter second 

17. We have also / = -20264 x i g 

and h g = 4-9348 / 
in any latitude and at any altitude. 

* At the level of the sea, in the latitude of London, g is 386-289 inches, and the 
corresponding length of the second pendulum is 39' 1393 inches, according to the 
determination of Major Kater. Conformably with this result are the numbers in 
the table following art. 30 of this section, computed at the expense of Messrs. Bra- 
mahs and Donkm, and obligingly communicated by them for this work. It has 
been suspected by M. Bessel, and demonstrated by Mr. Fra?icis Bailey, that, in 
the refined computations relative to the pendulum, the formulae for the reduction 
to a vacuum are inaccurate, and that, in consequence, we do not yet precisely 
know the length of a second pendulum. See Phil. Transac. 1832. 



248 DYNAMICS I PENDULUMS, &C. 

In other words, whatever be the force of gravity, the length 
of a second pendulum, and the space descended freely by a 
falling body in 1 second, are in a constant ratio. 

18. If /' be the length of a pendulum,^' the force of gravity, 
and t' the time of oscillation at any other place, then 



t: t 



"Sg'SS'' 



If the force of gravity be the same, 

t:t' ::•/: V/'. 

If the same pendulum be actuated by different gravitating 
forces, we have 



" \g yg' m 



* '•*'•'-' J-ih'-ss'-'Sg- 



When pendulums oscillate in equal times in different places, 
we have 

g-.g'y.l-.i: 

For the variations of gravity in different latitudes, see art. 9, 
pa. 238. 

18. If the arcs are not indefinitely short, let v denote the 
versed sine of the semi-arc of vibration ; then 



t = «Jj (i+^+ ¥ |s* 3 +&c.; 



In which, when the semi-arc of vibration does not exceed 4 
or 5 degrees, the third term of the series may be omitted. 

If the time of an oscillation in an indefinitely small arc be 
1 second, the augmentation of the time will be 

for a semi-arc of 30° 0*01675 

of 15° 0-00426 

of 10° 0*00190 

of 5° 0-00012 

of 2h° 0-00003 

So that for oscillations of 2h° on each side of the vertical, 
the augmentation would not occasion more than 2" difference 
in a day. 

19. If d denote the degrees in the semi-arc of an oscillating 
pendulum, the time lost in each second by vibrating in a cir- 

D 2 

cle instead of the cycloid, is ; and consequently the time 

lost in a whole day of 24 hours, or 24 x 60 x 60 seconds, is 



DYNAMICS, PENDULUMS, &C. 249 



| d 3 nearly. In like manner, the seconds lost per day by 
vibrating in the arc of A degree, is f A 2 . Therefore, if the 
pendulum keep true time in one of these arcs, the seconds 
lost or gained per day, by vibrating in the other, will be 
| (d 2 — A 3 ). So, for example, if a pendulum measure true 
time in an arc of 3 degrees, it will lose llf seconds a day by 
vibrating 4 degrees ; and 26f seconds a day by vibrating 5 de- 
grees : and so on. 

20. If a clock keep true time very nearly, the variation in 
the length of the pendulum, necessary to correct the error will 
be equal to twice the product of the length of the pendulum, 
and the error in time divided by the time of observation in 
which that error is accumulated. 

If the pendulum be one that should beat seconds, and /' 
the daily variation be given in minutes, and n be the number 
of threads in an inch of the screw which raises and depresses 

the bob of the pendulum, then A = = 

v 24 X 60 

•05434 n ?=■£•? n f nearly, for the number of threads which 
the bob must be raised or lowered, to make the pendulum vi- 
brate truly. 

21. For civil and military engineers, and other practical 
men, it is highly useful to have a portable pendulum, made 
of painted tape, with a brass bob at the end, so that the 
whole, except the bob, may be rolled up within a box, which 
may be enclosed in a shagreen case. The tape is marked 200, 
190, 180, 170, 160, &c. SO, 75, 70, 65, 60, at points which being 
assumed respectively as points of suspension, the pendulum 
will make 200, 190, &c. down to 60 vibrations in a minute. 
Such a portable pendulum may be readily employed in 
experiments relative to falling bodies, the velocity of sound, 
&c. The pendulum and its box may go in a waistcoat 
pocket. 

22. If the momentum of inertia (Def. 6) of a pendulum, 
whether simple or compound, be divided by the product of the 
pendulum's weight or mass into the distance of its centre of 
gravity from the point of suspension (or axis of motion), the 
quotient will express the distance of the centre of oscillation 
from the same point (or axis.) 

23. Whatever the number of separate masses or bodies which 
constitute a pendulum, it may be considered as a single 
pendulum, whose centre of gravity is at the distance d from 
the axis of suspension, or of rotation : then if k 2 denote 
the momentum of inertia of that body divided by its mass, 
the distance s o from the axis of rotation to the centre of 



250 DYNAMICS : PENDULUMS, &C. 

oscillation or the length of an equivalent simple pendulum, 
will be 

d 

24. To find the distance of the centre of oscillation from the 
point or axis of suspension, experimentally . Count the num- 
ber, n, of oscillations of the body in a very short arc in a minute ; 
then 

140850 
so = 

n % 

Thus, if a body so oscillating, made 50 vibrations in a minute : 

'' 140850 ^ nA . , 

then so = == 56*34 inches. 

2500 

Or, s o = 39i t 2 , in inches, t being the time of one oscillation 
in a very small arc. 

If the arc be of finite appreciable magnitude, the time of os- 
cillation must be reduced in the ratio 8-f-versin of semi-arc to 
8, before the rule is applied. 

25. From the foregoing principles are derived the following 
expressions for the distances of the centres of oscillation for the 
several figures, suspended by their vertices and vibrating flat- 
wise, viz. 

(1.) Right line, or very thin cylinder, s o=§ of its length. 

(2.) Isosceles triangle, s o=| of its altitude. 

(3.) Circle, s o=J radius. 

(4.) Common parabola, s o=^ of its altitude. 

2 m-\- 1 

(5.) Any parabola, so = xits altitude. 

v ' J r ' 3ra+l 

Bodies vibrating laterally or sideways, or in their own plane : 

(6.) In a circle, s o=| of diameter. 

(7.) In a rectangle suspended by one angle, s o = f of diago- 
nal. 

(8.) Parabola suspended by its vertex, so = f axis + ^ para- 
meter. 

(9.) Parabola suspended by middle of its base, s 0=4 axis + 5 

parameter. 

. c . , 3 arc X radius 

(10.) In a sector of a circle, s = 



(11.) In a cone, s = 4 axis-f 



4 chord 
(radius of base) 5 

5 axis 



* For some curious and valuable theorems, by Professor Airy, for the reduction 
of vibrations in the air to those in a vacuum, see Mr. F. Baily's paper referred to 
in the preceding note. 



DYNAMICS I CENTRE OP OSCILLATION. 251 

2 rad. 3 
(12.) In a sphere, so = rad. + d+ a , ? , — — tt 
v ' r ' 5 (d -\- rad.) 

Where e? is the length of the thread by which it is suspended. 
(13.) If the weight of the thread is to be taken into the ac- 
count, we have the following distance between the centre of 
the ball, and that of oscillation, where b is the weight of the 
ball, d the distance between the point of suspension and its 
centre, r the radius of the ball, and w the weight of the thread 
or wire,. 

(\ ?fl+f b) 4r 2 — 1 w(2 dr + d 2 ) .* , 

g o=- -— j\ r- % ! ■ ; or, if b be expressed 

(i w+b) d — rw r 

in terms of w considered as a unit, then a o — — - — -. 

B + 2 

(14.) If two weights w, w', be fixed at the extremities of a 
rod of given length w w', s being the centre of motion between 
w and w'; then, if d = s w, d = s w', and m the weight of an 
unit in length of the rod, we shall have 



s o 



m d 3 + 2 w d — m d 2 — 2wd 

the radii of the balls being supposed very small in comparison 

with the length of the rod. 

(15.) In the bob of a clock pendulum, supposing it two equal 

spheric segments joined at their bases, if the radii of those bases 

be each = g, the height of each segment v, and d the distance 

from the point of suspension to g the centre of the bob, then is 

e 4 +§ Z 2 v 2 -\-——v 4 

g = 3 d. — — : — J- 5 — ; which shows the distance of the 

g a +§ v z 

centre of oscillation below the centre of the bob. 

If r the radius of the sphere be known, the latter expression 

. -I I 2 v i r v — T V v 3 

becomes g o ==- r-, ; — P- — . 

d (r — f v) 

(16.) Let the length of a rectangle be denoted by /, its 

breadth by 2 iv, the distance (along the middle of the rectangle) 

from one end to the point of suspension by d, then the distance 

s o, from the point of suspension to the centre of oscillation, 

will be so= —j -= = il— rf+^n ~r , 

2 1 — d 2 1 — a 

whether the figure be a mere geometrical rectangle, or a pris- 
matic metallic plate of uniform density. 

It follows from this theorem that a plate of 1 foot long, and f- 
of a foot broad, and suspended at a fourth of a foot from either 
end, would vibrate as a half second pendulum. 

Also, that a plate a foot long, -^ of a foot wide, and suspended 
at £ of a foot from the middle, would vibrate 36,469 times in 5 
hours. 




252 DYNAMICS I CENTRE OF OSCILLATION. 

And hence the length of a foot may be determined ex 
perimentally by vibrations. 

(17.) If a thin rod, say of a foot in length, 
have a ball of an inch diameter at each end, a 
and b, and a moveable point of suspension, s ; 
then the time of oscillation of such a pendulum 
may be made as long as we please, by bringing 
the point of suspension nearer and nearer to the 
middle of the rod. 

Or, if the point of suspension be fixed the dis- 
tance s o (and consequently the time of oscilla- 
tions which is as ^s o) may be varied by placing 
A nearer or farther from s. And this is the 
principle of the metronome, by which musi- 
cians sometimes regulate their time. 

(18.) If the weight of the connecting rod be evanescent with 
regard to the weight of the balls a and b ; then if R=radius of 
the larger ball, r that of the smaller, d and d the distances of 
their respective centres from s : we shall have 

_ r 3 (5d 3 +2 R a ) + r 3 (5 d 2 +2r 2 ) 
S0 ~~ 5(BR 3 — dr 3 ) 

When r and r are equal, this becomes 

(19.) If the minor and major axes of an ellipse (or of an 
elliptical plate of wood or metal) be as 1 to -v/3, or as 1000 
to 1732 ; then if it be suspended at one extremity of the minor 
axis, the centre of oscillation will be at the other extremity of 
that axis, or its oscillations will be performed in the same time 
as those of a simple pendulum whose length is equal to the 
minor axis. 

The same ellipse also possesses this curious and useful pro- 
perty : viz. That any segment or any zone of the ellipse cut 
off by lines parallel to the major axis, whether it be taken 
near the upper part of the minor axis, near the middle, or near 
the bottom of the same, will vibrate in the same time as the 
whole ellipse, the point of suspension being at an extremity of 
the minor axis. 

26. It is evident, from art. 17, that pendulums in different 
latitudes require to be of different lengths, in order that they 
may perform their vibrations in the same time ; but besides 
this there is another irregularity in the motion of a pendulum 
in the same place, arising from the different degrees of tem- 
perature. Heat expanding, and cold contracting the rod of 



DYNAMICS I COMPENSATION PENDULUMS, &C. 253 

the pendulum, a certain small variation must necessarily fol- 
low in. the time of its vibrations ; to remedy which various 
methods have been invented for constructing what are com- 
monly called compensation pendulums, or such as shall always 
preserve the same distance between the centre of oscillation 
and the point of suspension ; but of these we shall describe 
two or three. 

Compound or Compensation Pendulums have received dif- 
ferent denominations, from their form and materials, as the 
gridiron pendulum, mercurial pendulum, &c. 

27. The Gridiron Pendulum consists of five rods of steel, 
and four of brass, placed in an alternate order, the middle rod 
being of steel, by which the pendulum ball is suspended ; these 
rods of brass and steel are placed in an alternate order, and so 
connected with each other at their ends, that while the ex- 
pansion of the steel rods has a tendency to lengthen the pen- 
dulum, the expansion of the brass rods acting upwards tends 
to shorten it. And thus, when the lengths of the brass and 
steel rods are duly proportioned, their expansions and contrac- 
tions will exactly balance and correct each other, and so pre- 
serve the pendulum invariably of the same length. 

Sometimes 3, 7, or 9 rods, are employed in the construction 
of the gridiron pendulum ; and zinc, silver, and other metals, 
may be used instead of brass and steel. 

28. The mercurial pendulum was invented by Mr. Graham, 
an eminent clockmaker, about the year 1715. Its rod was 
made of brass, and branched towards its lower end, so as to 
embrace a cylindric glass vessel 13 or 14 inches long, and 
about 2 inches diameter ; which being filled about 12 inches 
deep with mercury, forms the weight or ball of the pendulum. 
If, upon trial, the expansion of the rod be found too great for 
that of the mercury, more mercury must be poured into the 
vessel : if the expansion of the mercury exceeds that of the 
rod, so as to occasion the clock to go fast with heat, some 
mercury must be taken out of the vessel, so as to shorten the 
column. And thus may the expansion and contraction of the 
quicksilver in the glass be made exactly to balance the expan- 
sion and contraction of the pendulum rod, so as to preserve the 
distance of the centre of oscillation from the point of suspen- 
sion invariably the same. 

This kind of pendulum fell entirely into disuse soon after 
Graham's time ; but it has lately been re-adopted with con- 
siderable success by practical astronomers. A very instructive 
paper on its principles, construction, and use, has been pub- 
lished by Mr. F. Baily, in vol. i. part 2, Memoirs of the As- 
tronomical Society of London. 

34 Z 



254 DYNAMICS : COMPENSATION PENDULUMS. 

29. Reid's Compensation Pendulum is a recent 
invention of Mr. Adam Beid, of Woolwich, the con- 
struction of which is as follows : a n is a rod of wire, 
and z z a hollow tube of zinc, which slips on the wire, 
being stopped from falling off by a nut n, on which 
it rests ; and on the upper part of this cylinder of 
zinc rests the heavy ball b ; now the length of the 
tube z z being so adjusted to the length of the rod a n, 
that the expansions of the two bodies shall be equal 
with equal degrees of temperature ; that is, by making 
the length of the zinc tube to that of the wire, as the ex- 
pansion of wire is to that of zinc, it is obvious that the 
ball b will in all cases preserve the same distance from 
a ; for just so much as it would descend by the expan- 
sion of the wire downwards, so much will it ascend by the ex- 
pansion of the zinc upwards, and consequently its vibrations 
will in all temperatures be equal in equal times. 



30. Drummond's Compensation Pendulum. 

This was proposed by an artist in Lancashire more than 70 
years ago. A bar of the same metal with the rod of the pen- 
dulum, and of the same thickness and length, is placed against 
the back part of the clock-case : from the top of this a part 
projects, to which the upper part of the pendulum is connected 
by two fine pliable chains or silken strings, which just below 
pass between two plates of brass whose lower edges will al- 
ways terminate the length of the pendulum at the upper end. 
These plates are supported on a foot fixed to the back of the 
case. This bar rests upon an immoveable base on the lower 
part of the case, and is braced into a proper groove, which ad- 
mits of no motion any way but that of expansion and contract 
tion in length by heat and cold. In this construction, since 
the two bars are of equal magnitude and like constitution, their 
expansions and contractions will always be equal and in op- 
posite directions ; so that one will serve to correct and annihilate 
the effects of the other. 

An extensive and valuable table of the expansions of dif- 
ferent substances is given by Mr. Baily in the paper referred 
to above. 



DYNAMICS I COMPENSATION PENDULUMS. 



25; 



Table of Lengths and Vibrations of Pendulums. 

[See note at foot of page 250.] 



Length 


Time of 


No.Vibr. 


Length 


Time of 


No.Vibr. 


inches. 


vibration. 


per sec. 


inches. 


vibration. 


per sec. 


1-0 


0-1598 


6-256 


5-3 


0-3679 


2-717 


1-1 


0-1676 


5-965 


5-4 


03713 


2-692 


1-2 


0-1751 


5-711 


5-5 


0-3748 


2-667 


1-3 


0-1822 


5-487 


5-6 


0-3782 


2-643 


1-4 


0-1891 


5-287 


5-7 


0-3816 


2-620 


1-5 


0-1957 


5-108 


5-8 


0-3849 


2-597 


1-6 


0-2021 


4-945 


5-9 


0-3882 


2-575 


1-7 


0-2084 


4-798 


6-0 


0-3915 


2-554 


1-8 


0-2144 


4-663 


61 


0-3947 


2-533 


1-9 


0-2203 


4-538 


6-2 


0-3980 


2-512 


2-0 


0-2260 


4-423 


6-3 


0-4012 


2-492 


2-1 


0-2316 


4-317 


6-4 


0-4043 


2-472 


2-2 


0-2370 


4-217 








2-3 


0-2424 


4-125 


6-5 


0-4075 


2-453 


2-4 


0-2476 


4-038 


6-6 


0-4106 


2-435 


2-5 


0-2527 


3-956 


6-7 


0-4137 


2-416 


2-6 


0-2577 


3-879 


6-8 


0-4168 


2-399 


2-7 


0-2626 


3-807 


6-9 


0-4198 


2-381 


2-8 


0-2674 


3-738 


7-0 


0-4229 


2-364 


2-9 


0-2721 


3-673 


7-1 


0-4259 


2-347 


30 


0-2768 


3-612 


7-2 


0-4289 


2-331 


3-1 


0-2814 


3-553 


7-3 


0-4318 


2-315 


3-2 


0-2859 


3-497 


7-4 


0-4348 


2-399 


3-3 


0-2903 


3-443 


7-5 


0-4377 


2-284 


3-4 


0-2947 


3-392 


7.6 


0-4406 


2-269 


3-5 


0-2990 


3-344 


7-7 


0-4435 


2-254 


3-6 


0-3032 


3-297 


7-8 


0-4464 


2-240 


3-7 


0-3074 


3-252 


7-9 


0-4492 


2-225 


3-8 


0-3115 


3-209 


8-0 


0-4521 


2-211 


3-9 


0-3157 


3-167 


8-1 


0-4549 


2-198 


4-0 


0-3196 


3-128 


8-2 


0-4577 


2-184 


4-1 


0-3236 


3-089 


8-3 


0-4605 


2-171 


4-2 


0-3275 


3-052 


8-4 


0-4632 


2-158 


4-3 


0-3314 


3-016 


8-5 


0-4660 


2-145 


4-4 


0-3352 


2-982 


8-6 


0-4687 


2-133 


4-5 


0-3390 


2-949 


8-7 


0-4714 


2-121 


4-6 


0-3428 


2-916 


8-8 


0-4741 


2-108 


4-7 


0-3465 


2-885 


8-9 


0-4768 


2-097 


4-8 


0-3502 


2-855 


9-0 


0-4795 


2-085 


4-9 


0-3538 


2-826 


9-1 


0-4821 


2073 


5-0 


0-3574 


2-797 


9-2 


0-4848 


2-062 


5-1 


0-3609 


2-770 


9-3 


0-4874 


2-051 


5-2 


0-3644 


2-743 


9-4 


0-4900 


2-040 



256 



LENGTHS AND VIBRATIONS OF PENDULUMS. 



Length 


Time of 


No.Vibr. 


Length 


Time of 


No.Vibr. 


inches. 


vibration. 


per sec. 


inches. 


vibration. 


per sec. 


9-5 


0-4926 


2-029 


14-3 


0-6044 


1-654 


9-6 


0-4952 


2-019 


14-4 


0-6065 


1-648 


9-7 


0-4978 


2-008 


14-5 


0-6086 


1-642 


9-8 


0-5003 


1-998 


14-6 


0-6107 


1-637 


9-9 


0-5029 


1-988 


14-7 


0-6128 


1-631 


10-0 


0-5054 


1-978 


14-8 


0-6149 


1-626 


10-1 


0-5079 


1-968 


14-9 


0-6170 


1-620 


10-2 


0-5105 


1-958 


15-0 


0-6191 


1-615 


10-3 


0-5130 


1-949 


15-1 


0-6211 


1-609 


10-4 


0-5155 


1-939 


15-2 


0-6231 


1-604 


10-5 


0-5179 


1-930 


15-3 


0-6252 


1-599 


10-6 


0-5204 


1-921 


15-4 


0-6272 


1-594 


10-7 


0-5228 


1-912 


15-5 


0-6293 


1-589 


10-8 


0-5253 


1-903 


15-6 


0-6313 


1-584 


10-9 


0-5277 


1-894 


15-7 


0-6333 


1-579 


11-0 


0-5301 


1-886 


15-8 


0-6353 


1-574 


11-1 


0-5325 


1-877 


15-9 


0-6373 


1-569 


11-2 


0-5349 


1-869 


16-0 


0-6393 


1-564 


11-3 


0-5373 


1-861 


16-1 


0-6413 


1-559 


11-4 


0-5396 


1-853 


16-2 


0-6433 


1-554 


11-5 


0-5420 


1-845 


16-3 


0-6453 


1-549 


11-6 


0-5444 


1-837 


16-4 


0-6473 


1-544 


11-7 


0-5467 


1-829 


16-5 


0-6493 


1-539 


11-8 


0-5490 


1-821 


16-6 


0-6532 


1-535 


11-9 


0-5514 


1-813 


16-7 


0-6551 


1-531 


12-0 


0-5537 


1-806 


16-8 


0-6571 


1-526 


12-1 


0-5560 


1-798 


16-9 


0-6590 


1-521 


12-2 


0-5583 


1-791 


17-0 


0-6609 


1-517 


12-3 


0-5605 


1-783 


17-1 


0-6629 


1-512 


12-4 


0-5628 


1-776 


17-2 


0-6648 


1-508 


12-5 


0-5651 


1-769 


17-3 


0-6648 


1-504 


12-6 


0-5673 


1-762 


17-4 


0-6667 


1-499 


12-7 


0-5696 


1-755 


17-5 


0-6686 


1-495 


12-8 


0-5718 


1-748 


17-6 


0-6705 


1-491 


12-9 


0-5741 


1-741 


17-7 


0-6724 


1-487 


13-0 


0-5763 


1-735 


17-8 


0-6743 


1-482 


13-1 


0-5785 


1-728 


17-9 


0-6762 


1-478 


13-2 


0-5807 


1-721 


18-0 


0-6781 


1-474 


13-3 


0-5829 


1-715 


18-1 


0-6800 


1-470 


13-4 


0-5851 


1-709 


18-2 


0-6819 


1-466 


13-5 


0-5873 


1-703 


18-3 


0-6837 


1-462 


13-6 


0-5894 


1-696 


18-4 


0-6856 


1-458 


13-7 


0-5916 


1-690 


18-5 


0-6875 


1-454 


13-8 


0-5938 


1-684 


18-6 


0-6893 


1-450 


13-9 


0-5959 


1-678 


18-7 


0-6912 


1-446 


14-0 


0-5980 


1-672 


18-8 


0-6930 


1-442 


14-1 


0-6001 


1-666 


18-9 


0-6949 


1-439 


14-2 


0-6023 


1-660 


19-0 


0-6967 


1-435 



LENGTHS AND VIBRATIONS OF PENDULUMS. 



257 



Length 


Time of 


No.Vibr. 


Length 


Time of 


No.Vibr. 


inches. 


vibration. 


per sec. 


inches. 


vibration. 


per sec. 


19-1 


0-6985 


1-431 


23-9 


s 

0-7814 


1-279 


19-2 


0-7003 


1-427 


24-0 


0-7830 


1-277 


19-3 


0-7022 


1-424 


24-1 


0-7847 


1-274 


19-4 


0-7040 


1-420 


24-2 


0-7863 


1-271 


19-5 


0-7058 


1-416 


24-3 


0-7879 


1-269 


19-6 


0-7076 


1-413 


24-4 


0-7895 


1-266 


19-7 


0-7094 


1-409 


24-5 


0-7911, 


1-263 


19-8 


0-7112 


1-405 


24-6 


0-7927 


1-261 


19-9 


0-7130 


1-402 


24-7 


0-7944 


1-259 


20-0 


0-7148 


1-398 


24-8 


0-7960 


1-256 


20-1 


0-7166 


1-395 


24-9 


0-7976 


1-253 


20-2 


0-7184 


1-391 


25-0 


0-7992 


1-251 


20-3 


0-7201 


1-388 


25-1 


0-8008 


1-248 


20-4 


0-7219 


1-384 


25-2 


0-8024 


1-246 


20-5 


0-7237 


1-381 


25-3 


0-8040 


1-243 


20-6 


0-7254 


1-378 


25-4 


0-8056 


1-241 


20-7 


0-7272 


1-375 


25-5 


0-8071 


1-238 


20-8 


0-7289 


1-371 


25-6 


0-8087 


1-236 








25-7 


0-8103 


1-234 


20-9 


0-7307 


1-368 


25-8 


0-8119 


1-231 


21-0 


0-7324 


1-365 


25-9 


0-8134 


1-229 


21-1 


0-7342 


1-361 


26-0 


0-8150 


1-226 


21-2 


0-7359 


1-358 


26-1 


0-8166 


1-224 


21-3 


0-7377 


1-355 


26-2 


0-8181 


1.222 


21-4 


0-7394 


1-352 


26-3 


0-8197 


1-219 


21-5 


0-7411 


1-349 


26-4 


0-8212 


1-217 


21*6 


0-7428 


1-346 


26-5 


0-8228 


1-215 


21-7 


0-7446 


1-343 


26-6 


0-8244 


1-213 


21-8 


0-7463 


1-339 


26-7 


0-8259 


1-211 


21-9 


0-7480 


1-336 


26-8 


0-8275 


1-208 


22-0 


0-7497 


1-333 


26-9 


.0-8290 


1-206 


22-1 


0-7514 


1-330 


27-0 


0-8305 


1-204 


22-2 


0-7531 


1-327 


27-1 


0-8321 


1-201 


22-3 


0-7548 


1-324 


27-2 


0-8336 


1-199 


22-4 


0-7565 


1-321 


27-3 


0-8351 


1-197 


22-5 


0-7582 


1-318 


27-4 


0-8367 


1-195 


22-6 


0-7598 


1-315 


27-5 


0-8382 


1-193 


22-7 


0-7615 


1-313 


27-6 


0-8397 


1-191 


22-8 


0-7632 


1-310 


27-7 


0-8412 


1-189 


22-9 


0-7649 


1-307 


27-8 


0-8427 


1-186 


23-0 


0-7665 


1-304 


27-9 


0-8443 


1-184 


23-1 


0-7682 


1-301 


28-0 


0-8458 


1-182 


23-2 


0-7699 


1-298 


28-1 


0-8473 


1-180 


23-3 


0-7715 


1-296 


28-2 


0-8488 


1-178 


23-4 


0-7732 


1-293 


28-3 


0-8503 


1-176 


23-5 


0-7748 


1-290 


28-4 


0-8518 


1-173 


23-6 


0-7765 


1-287 


28-5 


0-8533 


1-171 


23-7 


0-7781 


1-285 


28-6 


0-8548 


1-169 


23-8 


0-7798 


1-282 


28-7 


0-8563 


1-167 



z 2 



258 



LENGTHS AND VIBRATIONS OF PENDULUMS. 



Length 


Time of 


No.Vibr. 


Length 


Time of 


No.Vibr. 


inches. 


vibration. 


per sec. 


inches. 


vibration. 


per sec. 


28-8 


0-8578 


1-165 


34-2 


0-9347 


1-069 


28-9 


0-8593 


1-163 


34-3 


0-9361 


1-068 


29-0 


0-8607 


1-161 


34-4 


0-9375 


1-066 


29-1 


0-8622 


1-159 


34-5 


0-9389 


1-065 


29-2 


0-8637 


1-157 


34-6 


0-9402 


1-063 


29-3 


0-8652 


1-155 


34-7 


0-9415 


1-062 


29-4 


0-8667 


1-154 


34-8 


0-9429 


1-060 


29-5 


0-8682 


1-152 


34-9 


0-9443 


1-059 


29-6 


0-8696 


1-150 


35-0 


0-9456 


1-057 


29-7 


0-8711 


1-148 


35-1 


0-9470 


1-055 


29-8 


0-8726 


- 1-146 


35-2 


0-9483 


1-054 


29-9 


0-8741 


1-144 


35-3 


0-9497 


1-052 


30-0 


0-8755 


1-142 


35-4 


0-9510 


1-051 


30-1 


0-8769 


1-140 


35-5 


0-9523 


1-050 


30-2 


0-8784 


1-138 


35-6 


0-9537 


1-048 


30-3 


0-8798 


1-136 


35-7 


0-9550 


1-047 


30-4 


0-8813 


1-135 


35-8 


0-9563 


1-045 


30-5 


0-8827 


1-133 


35-9 


0-9577 


1-044 


30-6 


0-8842 


1-131 


36-0 


0-9590 


1-042 


30-7 


0-8856 


1-129 


36-1 


0-9603 


1-041 


30-8 


0-8870 


1-127 


36-2 


0-9617 


1-039 


30-9 


0-8885 


1-125 


36-3 


0-9630 


1-038 


31-0 


0-8899 


1-123 


36-4 


0-9643 


1-036 


31-1 


0-8914 


1-121 


36-5 


0-9657 


1-035 


31-2 


0-8928 


1-120 


36-6 


0-9670 


1-034 


31-3 


0-8942 


1-118 


36-7 


0-9683 


1-032 


31-4 


0-8956 


1-116 


36-8 


0-9696 


1-031 


31-5 


0-8971 


1-114 


36-9 


0-9709 


1-029 


31-6 


0-8985 


1-112 


37-0 


0-9722 


1-028 


31-7 


0-8999 


1-111 


37-1 


0-9736 


1-027 


31-8 


0-9013 


1-109 


37-2 


0-9749 


1-025 


31-9 


0-9027 


1-107 


37-3 


0-9762 


1-023 


32-0 


0-9042 


1-105 


37-4 


0-9775 


1-024 


32-1 


0-9056 


1-104 


37-5 


0-9788 


1-021 


32-2 


0-9070 


1-102 


37-6 


0-9801 


1-020 


32-3 


0-9084 


1-100 


37-7 


0-9814 


1-018 


32-4 


0-9098 


1-099 


37-8 


0-9827 


1-017 


32-5 


0-9112 


1-097 


37-9 


0-9840 


1-016 


32-6 


0-9126 


1-095 


38-0 


0-9853 


1-014 


32-7 


0-9140 


1-094 


38-1 


0-9866 


1-013 


32-8 


0-9154 


1-092 


38-2 


0-9879 


1-012 


32-9 


0-9168 


1-090 


38-3 


0-9892 


1-010 


330 


0-9182 


1-089 


38-4 


0-9905 


1-009 


33- 1 


0-9196 


1-087 


38-5 


0-9918 


1-008 


33-2 


0-9210 


1-085 


88-6 


0-9931 


1-006 


33-3 


0-9224 


1-084 


38-7 


0-9943 


1-005 


33-4 


0-9237 


1-082 


38-8 


0-9956 


1-004 


33-5 


0-9251 


1-080 


38-9 


0-9969 


1-003 


33-6 


0-9265 


1-079 


39-0 


0-9982 


1-001 


33-7 


0-9279 


1-077 


39-1 


0-9995 


1-000 


33-8 


0-9293 


1-076 


39-2 


1-0001 


0-9993 


33-9 


0-9306 


1-074 








340 


0-9320 


1-072 








34-1 


0-9334 


1-071 









DYNAMICS : GYRATION AND ROTATION. 259 

Centre of Gyration, Principles of Rotation. 

1. The distance of r the centre of gyration, from c the centre 
or axis of motion, in some of the most useful cases, is exhibited 
below. 

In a circular wheel of uniform thickness c r = rad. ^/ §. 
In the periphery of a circle revolving \ __ , , 

about the diam 5 C R ~ rad * * 5 * 

In the plane of a circle .... ditto c r = \ rad. 

In the surface of a sphere . . ditto c r = rad. v/ f . 

In a solid sphere ditto c r = rad. \/ § 

= T 7 T r nearly. 
In a plane ring formed of circles whose } 3 , . 

radii are r, r, revolving about > c r = %/ 



> c r = 



centre ) 2 

In a cone revolving about its vertex . . c R = i y/ iztf+^r 2 

In a cone its axis . . c r - = r </ T 3 _. 

In a paraboloid c e = r y J. 

In a straight lever whose arms are r and r,CR = ^ —, — 7— r. 

2. If the matter in any gyrating body were actually to be 
placed as if in the centre of gyration, it ought either to be dis- 
posed in the circumference of a circle whose radius is c r, or at 
two points r, r', diametrically opposite, and at distances from 
the centre each = c r. 

3. By means of the theory of the centre of gyration, and the 
values of c r = §, thence deduced, the phenomena of rotation 
on a fixed axis become connected with those of accelerating 
forces : for then, if a weight or other moving power p act at a 
radius r to give rotation to a body, weight w, and dist. of centre 
of gyration from axis of motion = s, we shall have for the ac- 
celerating force, the expression 

J p r 3 + w g 3 
and consequently for the space descended by the actuating 
weight or power p, in a given time t, we shall have the usual 
formula 



- ,/ * <, « '-J7? 



&c. 



introducing the above value of f 

4. In the more complex cases, the distance of the centre of 
gyration from the axis of motion may best be computed from 
an experiment. Let motion be given to the system, turning 
upon a horizontal axis, by a weight p acting by a cord over a 



260 DYNAMICS t GYRATION AND ROTATION. 

pulley or wheel of radius r upon the same axis, and let s be the 
space through which the weight p descends in the time t, the 
proposed body whose weight is w turning upon the same axis 
with the same angular velocity : then 

I g p t 2 r 3 — 2 s p r % 

CR = s= >s|; 27^ • 

Example. — A body which weighs 100 lbs. turns upon a ho- 
rizontal axis, motion being communicated to it by a weight 
of 10 lbs. hanging from a very light wheel of 1 foot diameter. 
The weight descends 2 feet in 3 seconds. Required the dis- 
tance of the centre or circle of gyration from the axis of 
motion. 

Here, I take g = 32, instead of 32^, and obtain an approxi- 
mative result. Whence 

J 32 X 10 X 9 X ? - 4 X 10 X i [ 720 — 10 

4 X 100 ~"\ 400 

26-646 „ , 

= _i_ ^ 710 = ■■ = 1-3323 f. the answer. 

/i\J 

5. When the impulse communicated to a body is in a line 
passing through its centre of gravity, all the points of the body 
move forward with the same velocity, and in lines parallel 
to the direction of the impulse communicated. But when the 
direction of that impulse does not pass through the centre of 
gravity, the body acquires a rotation on an axis, and also a 
progressive motion, by which its centre of gravity is carried 
forward in the same straight line, and with the same velocity, 
as if the direction of the impulse had passed through the centre 
of gravity. 

The progressive and rotatory motion are independent of 
one another, each being the same as if the other had no ex- 
istence. 

6. When a body revolves on an axis, and a force is im- 
pressed, tending to make it revolve on another, it will revolve 
on neither, but on a line in the same plane with them, divid- 
ing the angle which they contain, so that the sines of the parts 
are in the inverse ratio of the angular velocities with which 
the body would have revolved about the said axis sepa- 
rately. 

7. A body may begin to revolve on any line as an axis that 
passes through its centre of gravity, but it will not continue to 
revolve permanently about that axis, unless the opposite rota- 
tory forces exactly balance one another. 

This admits of a simple experimental illustration. Suspend 
a thin circular plate of wood or metal by a cord tied to its 
edge, from a hook to which a rapid rotation can be given. 



DYNAMICS I AXES OF ROTATION. 261 

The plate will at first turn upon an axis which is in the con- 
tinuation of the cord of rotation. As the velocity augments, 
the plane will soon quit that axis, and revolve permanently 
upon a vertical axis passing through its centre of gravity, itself 
having assumed a horizontal position. 

The same will happen if a ring be suspended, and receive ro- 
tation in like manner. 

And if a flexible chain of small links be united at its two 
ends, tied to a cord and receive rotation, it will soon adjust itself 
so as to form a ring, and spin round in a horizontal plane. 

Also, if a flattened spheroid be suspended from any point, 
however remote from its minor axis, and have a rapid rotation 
given it, it will ultimately turn upon its shorter axis posited 
vertically. 

This evidently serves to confirm the motion of the earth upon 
its shorter axis. 

8. In every body, however irregular, there are three axes of 
permanent rotation, at right angles to one another. These are 
called the pri?icipal axes of rotation : they have this remark- 
able property, that the momentum of inertia with regard to any 
of them is either a maximum or a minimum. 

Central Forces. 

Def. 1. Centripetal force is a force which tends constantly 
to solicit or to impel a body towards a certain fixed point or 
centre. 

2. Centrifugal force is that by which it would recede from 
such a centre, were it not prevented by the centripetal force. 

3. These two forces are, jointly, called central forces. 

4. When a body describes a circle by means of a force direct- 
ed to its centre, its actual velocity is everywhere equal to that 
which it would acquire in falling by the same uniform force 
through half the radius. 

5. This velocity is the same as that which a second body 
would acquire by falling through half the radius, whilst the 
first describes a portion of the circumference equal to the whole 
radius. 

6. In equal circles the forces are as the squares of the times 
inversely. 

7. If the times are equal, the velocities are as the radii, and 
the forces are also as the radii. 

8. In general, the forces are as the distances or radii of the 
circles directly, and the squares of the times inversely. 

9. The squares of the times are as the distances directly, and 
the forces inversely. 

35 



262 DYNAMICS I CENTRAL FORCES. 

10. Hence, if the forces are inversely as the squares of the 
distances, the squares of the times are as the cubes of the dis- 
tances. That is, 

if p :/ :: d* : d 2 , then t 2 : t 2 :: d 3 : d\ 

11. The right line that joins a revolving body and its centre 
of attraction, called the radius vector, always describes equal 
areas in equal times, and the velocity of the body is inversely 
as the perpendicular drawn from the centre of attraction to the 
tangent of the curve at the place of the revolving body. 

12. If a body revolve in an elliptic orbit by a force directed 
to one of the foci, the force is inversely as the square of the 
distance : and the mean distances and the periodic times have 
the same relation as in art. 10. This comprehends the case of 
the planetary motions. 

13. If the force which retains a body in a curve increase in 
the simple ratio as the distance increases, the body will still 
describe an ellipse ; but the force will in this case be directed 
to the centre of the ellipse ; and the body in each revolution 
will twice approach towards it, and again twice recede from 
that point. 

14. On the principles of central forces depend the operation 
of a conical pendulum applied as a governor or regulator to 
steam engines, water mills, &c. 

This contrivance will be readily comprehended from the 
marginal figure, where a a is a vertical shaft capable of turn- 
ing freely upon the sole a. c d, c p, are two bars which move 
freely upon the centre c, and carry 
at their lower extremities two equal 
weights p, q ; the bars c d, c f, are 
united, by a proper articulation, to 
the bars g, h, which latter are at- 
tached to a ring i, capable of sliding 
up and down the vertical shaft a a. 
When this shaft and connected ap- 
paratus are made to revolve, in vir- 
tue of the centrifugal force, the balls 
p q fly out more and more from a a, 
as the rotatory velocity increases ; 

if, on the contrary, the rotatory velocity slackens, the balls 
descend and approach a a. The ring i ascends in the former 
case, descends in the latter : and a lever connected with i may 
be made to correct appropriately the energy of the moving 
power. Thus, in the steam engine, the ring may be made to 
act on the valve by which the steam is admitted into the cylin- 
der ; to augment its opening when the motion is slackening, 
and reciprocally diminish it when the motion is accelerated. 




DYNAMICS I GOVERNOR. 263 

The construction is, often, so modified that the flying out of 
the balls causes the ring i to be depressed, and vice versa ; but 
the general principle is the same. 

Here, if the vertical distance of p or q below c, be denoted by 
d, the time of one rotation of the regulator by t, and 3-141593 
by 7t, the theory of central forces gives 

t=2 7t — - = 1*10784 s/ d. 



-■ J4- 



Hence, the periodic time varies as the square root of the alti- 
tude of the conic pendulum, let the radius of the base be what 
it may. Also, when i c q=i c p=45°, the centrifugal force of 
each ball is equal to its weight. 



Inquiries connected with Rotation and Central Forces. 

1. Suppose the diameter of a grindstone to be 44 inches, and 
its weight half a ton ; suppose also that it makes 326 revolu- 
tions in a minute. What will be the centrifugal force, or its 
tendency to burst ? 

Here , - Uj^JV^^l^JL _ 47 . 22 w = 23 . 6 tons , 

the measure of the required tendency. 

2. If a fly wheel 12 feet diameter, and 3 tons in weight, re- 
volve in 8 seconds : and another of the same weight revolves 
in 6 seconds : what must be the diameter of the last, when their 
centrifugal force is the same ? 

By art. 8, Central Forces, f ;/:: — :— . Therefore, since f 

- D d , d t 2 12X36 cs * 
is = /,—- = —, or d — = = 6| feet, the answer. 

J T 2 t 2 T 3 64 

3. If a fly of 12 feet diameter revolve in 8 seconds, and another 
of the same diameter in 6 seconds : what is the ratio of their 
weights when their central forces are equal ? 

By art. 6, Central Forces, the forces are as the squares of 
the times inversely when the weights are equal : therefore when 
the weights are unequal, they must be directly as the squares of 
the times, that the central forces may be equal. 
Hence w:\v :: 36 : 64:: 1 :1|. 
That is, the weight of the more rapidly to that of the more 
slowly revolving fly, must be as 1 to 1-J, in the case proposed. 



264 dynamics: fly-wheels. 

4. If a fly 2 tons weight and 16 feet diameter, is sufficient 
to regulate an engine when it revolves in 4 seconds ; what 
must be the weight of another fly of 12 feet diameter re- 
volving in 2 seconds, so that it may have the same power upon 
the engine ? 

Here, by art. 8, Central Forces, we must have — — = ; 

., c wd/ 2 40cwt. X16X4 160 . _. . . 

therefore w = — = — — = = = 13$ cwt, the 

d t 3 12x16 12 

weight of the smaller fly.* 

Note. — A fly should always be made to move rapidly. If it 

be intended for a mere regulator, it should be near the first 

mover. If it be intended to accumulate force in the working 

point, it must not be far separated from it. 

5. Given the radius r of a wheel, and the radius r of its axle, 
the weight of both, w, and the distance of the centre of gyration 
from the axis of motion, ? ; also a given power p acting at 
the circumference of the wheel ; to find the weight w raised 
by a cord folding about the axle, so that its momentum shall be 
a maximum. 

Here w = 

n /(r 4 p 2 4-2 n 2 F^w-\-g 4 w 2 -{- vw Rre 3 +p 3 r 3 r) — r 3 p — fiv. 

_ 

Cor. 1. When r = r, as in the case of the single fixed pul- 

e 4 
ley then w = \/(2 p 3 R 3 -f 2 r p g 3 w + — w* -f p iv r g 3 ) — 

R 

\W—F. 
R 2 

Cor. 2. When the pulley is a cylinder of uniform matter 
^ = $ r 2 ? and the express, becomes w = \/[r 3 (2 p 3 -f | p w 
+ i IV 2 )] — i w—v. 

6. Let a given power p be applied to the circumference of a 
wheel, its radius r, to raise a weight w at its axle, whose radius 
is r, it is required to find the ratio of r and r, when w is raised 
with the greatest momentum ; the characters w and g denoting 
the same as in the last proposition. 

Here r = ^[p 2 w 3 + p 3 to + w)] p w 
p {q + w) 

F V 7 V 2 W V 7 V) . 

* Since — = — ; therefore r = =7^— nearly ; when the weight w, radius 

w g r gr 32 r 

r, and velocity v, are given. 

4 7t 2 r w 1-2273 r 

If r and t, the time, are given, then r = ——=:—-— w. r torn one or 

g P P 

other of these f the force may be found. 



DYNAMICS I INQUIRIES IN ROTATION. 265 

Cor. When the inertia of the machine is evanescent, with 

respect to that of p + w, then is r = r </{\-\ ) — 1. 

w 

7. If any machine whose motion accelerates, the weight will 
be moved with the greatest velocity when the velocity of the 

p 
power is to that of the weight as 1+p s/(l-\ ) to 1 ; the in 

ertia of the machine being disregarded. 

8. If in any machine whose motion accelerates, the descent 
of one weight causes another to ascend, and the descending 
weight be given, the operation being supposed continually re- 
peated, the effect will be greatest in a given time when the as- 
cending weight is to the descending weight as 1 to 1*618, in 
the case of equal heights ; and in other cases when it is to the 
exact counterpoise in a ratio which is always between 1 to 1§ 
and 1 to 2. 

9. The following general proposition with regard to rotatory 
motion will be of use in the more recondite cases. 

If a system of bodies be connected together and supported at 
any point which is not the centre of gravity, and then left to 
descend by that part of their weight which is not supported, 
2 g multiplied into the sum of all the products of each body into 
the space it has perpendicularly descended, will be equal to the 
sum of all the products of each body into the square of its ve- 
locity. 

Percussion or Collision. 

1. Defs. In the ordinary theory of percussion, or collision, 
bodies are regarded as either hard, soft, or elastic. A hard 
body is that whose parts do not yield to any stroke or percus- 
sion, but retains its figure unaltered. A soft body is that whose 
parts yield to any stroke or impression, without restoring them- 
selves again, the shape of the body remaining altered. An elas- 
tic body is that whose parts yield to any stroke, but presently 
restore themselves again, so that the body regains the same 
figure as before the stroke. When bodies which have been sub- 
jected to a stroke or a pressure return only in part to their 
original form, the elasticity is then imperfect : but if they re- 
store themselves entirely to their primitive shape, and employ 
just as much time in the restoration as was occupied in the 
compression, then is the elasticity perfect. 

It has been customary to treat only of the collision of bodies 
perfectly hard or perfectly elastic : but as there do not exist 
in nature any bodies (which we know) of either the one or the 

2 A 



266 DYNAMICS : COLLISION OP BODIES. 

other of these kinds, the usual theories are but of little service 
in practical mechanics, except as they may suggest an extension 
to the actual circumstances of nature and art. 

2. The general principle for determining the motions of bo- 
dies from percussion, and which belongs equally to both elastic 
and non-elastic bodies, is this : viz. that there exists in the bo- 
dies the same momentum, or quantity of motion, estimated in 
any one and the same direction, both before the stroke and after 
it. And this principle is the immediate result of the law of 
nature or motion ; that reaction is equal to action, and in a con- 
trary direction ; from whence it happens, that whatever motion 
is communicated to one body by the action of another, exactly 
the same motion does this latter lose in the same direction, or 
exactly the same does the former communicate to the latter in 
the contrary direction. 

From this general principle too it results, that no alteration 
takes place in the common centre of gravity of bodies by their 
actions upon one another ; but that the said common centre of 
gravity perseveres in the same state, whether of rest or of uni- 
form motion, both before and after the impact. 

3. If the impact of two perfectly hard bodies be direct, they 
will, after impact, either remain at rest, or move on uniformly 
together with different velocities, according to the circumstances 
under which they met. 

Let b and b represent two perfectly hard bodies, and let the 
velocity of b be represented by v, and that of b by v f which 
may be taken either positive or negative, according as b moves 
in the same direction as b, or contrary to that direction, and it 
will be zero when b is at rest. This notation being understood, 
all the circumstances of the motions of the two bodies, after 
collision, will be expressed by the formula : 

. », b v± b c 

velocity a= ■ — z — , 

J b -f- b ' 

which being accommodated to the three circumstances under 
which v may enter become 

lor'tvsr B y + b v Cwhen both bodies moved in 
^ = b + b \ the same direction 

b v — b v c when the bodies moved in 



velocity = ■ — r 

J b -f b 



contrary directions 



velocitv = B + b S when the hod y b was 
b v c a t rest. 

These formulae arise from the supposition of the bodies 

being perfectly hard, and consequently that the two after 

impact move on uniformly together as one mass. In cases 



dynamics: collision of bodies. 267 

of perfectly elastic bodies, other formulae have place which ex- 
press the motion of each body separately ; as in the following 
proposition. 

4. If the impact of two perfectly elastic bodies be direct, 
their relative velocities will be the same both before and after 
impact, or they will recede from each other with the same 
velocity with which they met ; that is, they will be equally 
distant, in equal times, both before and after their collision, 
although the absolute velocity of each may be changed. The 
circumstances attending this change of motion in the two 
bodies, using the above notation, are expressed in the two fol- 
lowing formulae : 

2 b v + (b — b) v , , ., c 

—^-, — 7 - — the velocity of b 

b + o 

2bv + (b — b) v , , . _„ 

; — r the velocity of b 

b + b J 

which needs no modification, when the motion of b is in the 

same direction with that of b. 

5. In the other case of b's motion, the general formulae be- 
come 

— 2 b v + (b — b) v . . . ± B 
—t — the velocity ot b 

2bv — (b — b) v . , 

— the velocity ot o 



b + b 

when b moves in a contrary direction to that of b, which arises 

from taking v negative. And 

(b — b) v 

ir-r- the velocity of b 

2 B v 

— — r the velocity ot o 

when b was at rest before impact, that is, when v = 0. 

6. If a perfectly hard body b, impinge obliquely upon a per- 
fectly hard and immoveable plane A d, it will after collision 
move along the plane in the direction c a. 

And its velocity before impact 

Is to its velocity after impact 

As radius 

Is to the cosine of the angle bcd. 
But if the body be elastic, it will re- 
bound from the plane in the direction 
c e, with the same velocity, and at the 
same angle with which it met it, that is, 
the angle ace will be equal to the an- 
gle BCD. 




268 



DYNAMICS I COLLISION OF BODIES. 



7. In the case of direct impact, if b be the striking body, h 
the body struck, v and v their respective velocities before im- 
pact, u and u their velocities afterwards ; then the two follow- 
ing are general formulae 



viz. 



u = v 



n 



\b -i- bi 



<B + b 

/v — v\ 
u == v + n( — — )b 
\b + b' 

In these, if n = 1, they serve for non-elastic bodies ; if n =» 
2, for bodies perfectly elastic. If the bodies be imperfectly 
elastic, n has some intermediate value. 

When the body struck is at rest, the preceding equations 
become, 



u = v 



nvb 



u = 



n v b 



n — 



u (b + b) 
B + b B + b BV 

from which the value of n may be determined experimentally. 

8. In the usual apparatus for experiments on collision, balls 
of different sizes and of various substances are hung from differ- 
ent points of suspension on a horizontal bar. h r m n is an arc 
of a circle whose centre is s ; and its graduations, 1, 2, 3, 4, 5, &c. 
indicate the lengths of cords, as 
measured from the lowest point. 
Any ball, therefore, as p, may be 
drawn from the vertical, and 
made to strike another ball hang- 
ing at the lowest point, with any 
assigned velocities, the height to 
which the ball struck ascends on 
the side a m furnishes a measure 
of its velocity ; and from that the 
value of n may be found from the 
last equation. Balls not required in an individual experiment 
may be put behind the frame, as shown at a and b. 

The cup c may be attached to a cord, and carry a ball of 
clay, &c. when required. 

Example. — Suppose that a ball weighing 4 ounces strikes 
another ball of the same substance weighing 3 ounces, with a 
velocity of 10, and communicates to it a velocity of 8i : what, 
in that case, will be the value of n ? 

u (b + b) 8U7 571 




Here n 



b v 



4 x 10 



40 



= 1-44375, the 



index of the degree of elasticity ; perfect elasticity being indi- 
cated by 2. 



DYNAMICS : PRINCIPLES OF CHRONOMETERS. 269 



Principles of Chronometers. 

1. Clockwork, regulated by a simple balance, is inadequate 
to the accurate mensuration of time. 

2. Clockwork, regulated by a pendulum vibrating in the arch 
of a circle, is of itself inadequate to the accurate mensuration of 
time. 

1st. Because the vibrations in greater or smaller arches are 
not performed in equal times. 2dly. Because the length of 
the pendulum is varied by heat and cold. 

3. Clockwork, regulated by a pendulum vibrating in the 
arch of a cycloid, is inadequate to the accurate mensuration of 
time. 

The isochronism of the vibrations of a cycloidal pendulum 
in greater and smaller arches is true only on the hypothesis, 
that the pendulum moves in a non-resisting medium, and that 
the whole mass of the pendulum is concentrated in a point, 
both of which positions are false. For these reasons the 
application of the cycloid in practice has been entirely relin- 
quished. 

4. Modern time-keepers owe almost the whole of their su- 
periority over those formerly made to two things ; 1st, the ap- 
plication of a thermometer ; 2dly, the particular construction 
of the escapement. 

5. Metals expand by heat and contract by cold. This is 
proved experimentally by the pyrometer. Metallic bars of 
the same kind are found to expand in proportion to their length. 
Metals of different kinds expand in different proportions : thus 
the expansion of iron and steel are as 3, copper 4§, brass 5, tin 
6, lead 7. Hence pendulum rods, expanding and contracting 
by the successive changes of temperature, affect the going of 
the clocks to which they are applied. 

Various have been the contrivances to correct the errors of 
pendulums from their contraction and expansion by heat and 
cold ; the principal of these are described under the subject of 
pendulums (p. 253). 

6. The balance of a watch is analogous to the pendulum in 
its properties and use. 

The simple balance is a circular annulus, equally heavy in 
all its parts, and concentrical with the pivots of the axis on 
which it is mounted. This balance is moved by a spiral 
spring called the balance-spring, the invention of the ingenious 
Mr. Hook. 

7. The pendulum requires a less maintaining power than the 
balance. . 

36 2 A 2 



270 DYNAMICS : PRINCIPLES OF CHRONOMETERS. 

Hence the natural isochronism of the pendulum is less dis- 
turbed by the relatively small inequalities of the maintaining 
power. 

8. The spring's elastic force which impels the circumference 
of the balance, is directly as the tension of the spring ; that is, 
the weights necessary to counterpoise a spiral spring's elastic 
force, when the balance is wound to different distances from 
the quiescent point, are in the direct ratio of the arcs through 
which it is wound. 

9. The vibrations of a balance, whether through great or 
small arches, are performed in the same time. 

For the accelerating force is directly as the distance from 
the point of quiescence ; hence, therefore, the motion of the 
balance is analogous to that of a pendulum, vibrating in cy- 
cloidal arches. 

10. The time of the vibration of a balance is the same as if 
a quantity of matter, whose inertia is equal to that by which 
the mass contained in the balance opposes the communica- 
tion of motion to the circumference, described a cycloid 
whose length is equal to the arc of vibration described by the 
circumference, the accelerating force being equal to that of the 
balance. 

Because in both cases the spaces described would be equal, 
as also the accelerating forces in corresponding points, and 
therefore the times of description. 

11. If 1 denote the accelerating force of gravity, l the 
length of a pendulum vibrating seconds in a cycloid, a the 
semi-arc of vibration of the balance, t the time of vibration, 
and f the accelerating force of the balance, then will t = 



J 



L X F 



12. Let \ g be the space which a body falling freely from 
a state of rest describes in 1", and p = 3-141593 the circum- 
ference of a circle whose diameter is unity, then will 



-J 



fa 
gv' 



In this expression for the time of vibration, the letter a de- 
notes the length of the semi-arc of vibration ; if this arc 
should be expressed by a number of degrees, c°, and r be the 

radius of the balance, then a will be = ^ ; and this quan- 

180 

tity being substituted for a, the time of a vibration will be t = 



J. 



1 p*c° 


gFX 180° 


\p c r 



DYNAMICS : PRINCIPLES OF CHRONOMETERS. 271 

; let the given arc be 90°, in this case t = 



13. If the spring's elastic force, when wound through the 
given angle or arc a= 90° from the quiescent position, be = 
p ; the weight of the balance, and the parts which vibrate 
with it = w, the distance of the centre of gyration from the 



axis of motion §, then will t 



\wp 3 z 2 
\2prg 



These are expressions for the time of a vibration, whatever 
may be the figure of the balance, the other conditions remain- 
ing the same as above stated. If the balance be an annulus or 

- v 

a cylindrical plate, g = — — , and the time of vibration t = 

v 2 



J 



w p 3 r 



4p# 

14. The times of vibration of different balances are in a 
ratio compounded of the direct subduplicate ratios of their 
weights and semidiameters, and the inverse subduplicate ratio 
of the tensions of the springs or of the weights which counter- 
poise them, when wound through a given angle. 

15. The times of vibration of different balances are in a 
ratio compounded of the direct simple ratio of the radii, and 
direct subduplicate ratio of their weights, and the inverse 
subduplicate ratio of the absolute forces of the springs at a 
given tension. 

16. Hence the absolute force of the balance spring, the 
diameter and weight of the balance being the same, is inversely 
as the square of the time of one vibration. 

17. The absolute force or strength of the balance spring, the 
time of one vibration, and the weight of the balance being the 
same, is as the square of the diameter and the balance. 

18. The weight of the balance, the strength of the spring 
and time of vibration being the same, is inversely as the square 
of the diameter. 

Hence a large balance vibrating in the same time, with the 
same spring, will be much lighter than a small one. 

19. If the rim of the balance be always of the same breadth 
and thickness, so that the weight shall be as the radius, the 



272 DYNAMICS I PRINCIPLES OP CHRONOMETERS. 

strength of the spring must be as the cube of the diameter of 
the balance, that the time of vibration may continue the 
same. 

20. If a balance be made with two balls joined by a rod, and 
the weights and distances of these balls from their common 
centre of motion be unequal, but such that each separately 
would vibrate in the same time ; the centre of gravity of these 
balls will not coincide with their centre of motion, nor will 
they poise each other. 

21. The momentum of the balance is increased better by 
increasing its diameter than its weight. 

23. A stronger balance-spring is preferable to a weaker. 
Because the force of this spring upon the balance remain- 
ing the same, whilst the disturbing force varies, the errors 
arising from the variation will be less, as the fixed force is 
greater. 

23. The longer a detached balance continues its motion the 
better. 

Because, 1st. The friction in this case is less, and therefore 
the natural isochronism of the vibration is less disturbed. 
2dly. When applied to the watch, it requires a less maintain- 
ing power, and therefore the variations in the intensity of the 
maintaining power will be less. 3dly. The maintaining power 
being less, the friction of the wheel-work will be less, and 
therefore the motion more regular. 4thly. The pressure on the 
escapement will be less, and therefore the oscillations of the 
balance less disturbed. 

24. The greater is the number of vibrations performed by 
a balance in a given time, the less susceptible is it of external 
agitations. 

25. Slow vibrations are preferable to quick vibrations : but 
there is a limit ; for if the vibrations be too slow, the watch 
will be liable to stop. 

If we regarded only the effect of external agitations, balances 
that vibrate quick should be preferred to such as vibrate slow ; 
but they are attended with two inconveniences, greater than 
that which we would avoid. 1st. In two balances of the 
same weight and diameter, the friction on the pivots increases 
with the number of vibrations. 2dly. It appears by ex- 
perience that the motion of the same detached balance con- 
tinues longer, when its vibrations are slow, than when they are 
quick. 

26. A balance should describe as large arches as possible, as 
suppose 240°, 260°, 300°, or an entire circle. 

First, because the momentum of the balance is thus in- 



DYNAMICS I PRINCIPLES OF CHRONOMETERS. 273 

creased ;, and therefore the inequalities in the force of the 
maintaining power bear a less proportion to it, and of conse- 
quence will have less influence. 2dly. The balance is less sus- 
ceptible of external agitations. 3dly. A given variation in the 
extent of the vibrations produces a less variation in the going 
of the machine. But care must be taken, that in these great 
vibrations, the spring shall neither touch any obstacle, nor its 
spires touch each other in contracting. 

27. The times of vibration in larger arches are sometimes 
shorter, sometimes longer, than in less arches, 

28. A uniform spiral spring may be rendered perfectly iso- 
chronal, by adjusting its length and number of spires. 

This is the opinion of Mr. Berthoud. His reasoning seems 
to be this r if the spring forming a spiral of a certain species 
be so disposed, that when wound through different angles, the 
accelerating elastic forces of the spires, from the centre to- 
wards the circumference, increase faster than they ought to do 
in order to render the vibrations isochronal, it may be other- 
wise so disposed, namely, by making the spires approach more 
nearly to equality with each other in succession, that the law 
shall vary in such a manner as absolute isochronism requires. 
But as the fundamental property of springs, namely, that as 
the tension is, so is the force, is determined by experiment, 
so must this property likewise be ascertained in the same 
manner. Accordingly Berthoud tells us, that having attached 
to a balance a spiral of very large folds, making but three 
turns, and whose diameter was 15 lines, the angles through 
which it was wound being successively 5°, 10°, 15°, 20°, 25°, 
30°, 35°, 40°, 45°, 60°, 120°, the counterpoising weights in 
grains were 10£, 21, 32, 42, 54, 65, 76, 88, 99, 134, 278. 
The same spring forming very small spires, making five 
turns in eight lines diameter, the angles through which it 
was wound being the same as before, the counterpoising 
weights were 11, 22, 33, 45, 56, 67, 78, 89, 100, 133,250 
grains. These experiments, he tells us, were made with great 
care ; and they show that the same spiral, its length continuing 
unchanged, when folded in large and small spires, has a suf- 
ficient difference in its progression to vary its isochronism : 
when folded in large spires, according to the first experiment, 
the vibrations in larger arcs are accelerated ; and by the second 
experiment, when folded in narrow spires, they are rendered 
slower. 

29. A spiral spring may be rendered isochronal by a 
proper adjustment of its strength and thickness in different parts. 



274 DYNAMICS I CHRONOMETERS. 

30. A spiral spring which is not isochronal, may be rendered 
such by the addition of two auxiliary springs, whose points of 
quiescence are properly adjusted. 

This was the ingenious invention of Mr. Mudge ; the theory 
of which construction is delivered in the Phil. Trans, for the 
year 1794, by Mr. Atwood. 

31. The influence of the maintaining power on the balance, 
in restoring the motion which it loses by friction, or other- 
wise, may be either constant or interrupted. 

This depends on the escapement ; when the action of the 
maintaining power is constant, the escapement is called either 
the recoil or the dead-beat ; when it is interrupted, the escape- 
ment is said to be detached. 

32. By escapement is understood the means by which the 
action of the wheels is applied to maintain the vibration of the 
balance ; and it consists of the balance-wheel and pallets. 

33. Pallets are small plates or levers attached to the axis or 
verge of the balance, which received the impulse of the balance- 
wheel produced by the maintaining power, and thus continually 
renew the motion which the balance loses by friction, or other 
resistance. 

In a recoil escapement, when one tooth of the balance-wheel 
drops off the first pallet, the other acting tooth falls on the in- 
clined plane of the other pallet, which, meeting it obliquely, 
causes the balance-wheel to recoil, from which circumstance 
this escapement derives its name. 

In the dead-beat escapement, when one tooth of the balance- 
wheel drops off the inclined plane of the first pallet, the other 
acting tooth immediately falls upon the convex surface of the 
other pallet, which surface being concentrical with the axis of 
the balance, the wheel continues at rest until, by the motion 
of the pallet or cylinder, the inclined plane of the tooth comes 
to act upon the face of this latter pallet or edge of the cylin- 
der, which then, by its pressure on that edge, throws the cy- 
linder round, and thus gives motion to the balance ; then 
instantly entering the cavity of the cylinder, it falls upon the 
concave surface, and for the same reason as before continues at 
rest, until the balance spring drives the cylinder round in a con- 
trary direction to what it did before, so as that the inclined plane 
of the tooth may act on the second edge of the cylinder ; which 
pressure throws the cylinder round in the contrary direction, 
and the tooth gets out of the cavity, and at that instant the sub- 
sequent tooth falls upon the convex surface, and so on. From 
the quiescence of the balance-wheel during the interval of time 



DYNAMICS : ESCAPEMENTS. 275 

that elapses between the falling of the acting tooth on the sur- 
face and its pressure on the edge of the cylinder, this escape- 
ment is called the dead-beat. 

In the detached escapement the motion of the maintaining 
power is suspended during almost the whole time of vibration ; 
just at the end of the return of the balance it unlocks the wheel- 
work, and a tooth of the balance-wheel, immediately acting on 
the pallet, restores the motion which the balance had lost ; and 
having given its impulse, the wheel-work is instantly locked 
again, and the balance performs its vibration freely and dis- 
engaged from all other parts of the machine. 

34. In the escapement of recoil, the vibrations are quicker 
than if the balance or pendulum vibrated freely. 

For the recoil shortens the ascending part of the vibration by 
contracting the extent of the arc : and the reaction of the wheel 
accelerates the descending part of the vibration. 

35. In the dead-beat escapement, the vibrations are slower 
than when they are performed in a detached state. 

For the pressure of the tooth on the surface of the cylinder 
retards that part of the vibration which is performed while the 
cylinder, by the motion of the balance spring, revolves so far 
as to bring the tooth to the edge of the cylinder : and if the 
maintaining power be increased, the pressure of the tooth on 
the cylinder may become so great as entirely to stop the motion. 
When the tooth has communicated its impulse to the edge of 
the cylinder, it moves almost freely ; and as the tooth does not 
yet press with its entire force on the next surface, the cylinder 
will indeed describe a larger arc, and therefore on that account 
the time may be shortened ; but when it has consumed all the 
impulse of the wheel, it returns by the sole force of elasticity ; 
now the pressure of the tooth causes a friction which diminishes 
the tendency to return to the point of rest, so that the balance 
performs its vibrations slower. 

36. In the escapement of recoil, if the maintaining power be 
increased, the vibrations will be performed in larger arches, but 
in less time. 

Because the greater pressure of the crown wheel on the pal- 
let will cause the balance to vibrate through larger arches ; and 
the time, on this account, will be less increased, than it will 
be diminished by the acceleration of the balance by that pres- 
sure, and the diminution of the time of recoil. 

37. In the escapement of the cylinder or dead-beat, an in- 
crease of the maintaining power renders the vibrations larger, 
and at the same time slower. 

Because the greater pressure of the tooth on the edge of 
the cylinder throws it round through a greater arch ; and its 



276 



DYNAMICS : ESCAPEMENTS. 



increased pressure on both surfaces of the cylinder retards its 
motion. 



The upper part of the first marginal dia- 
gram exhibits the anchor recoil ; the lower 
Graham's dead-beat escapement. 




The second diagram represents Mr. Ar- 
nold's watch escapement. The pin a, pro- 
jecting from the verge or axis of the balance, 
moving towards b, carries before it the 
spring b, and with it the stiffer spring c, so 
as to set at liberty the tooth d, which rests 
on a pallet projecting from the spring. The 
angle e of the principal pallet has then just 
passed the tooth f, and is impelled by it 
until the tooth g arrives at the detent. In 
the return of the balance, the pin a passes 
easily by the detent, by forcing back the spring b. The screw 
h serves to adjust the position of the detent, which presses 
against it. 

38. The escapement can render those vibrations only iso- 
chronal, whose inequality proceeds from the maintaining power, 
and not such as are produced by external agitations. 

39. The effect of external agitations on the balance may be 
counteracted by the double escapement. 

In this escapement, two equal balances are so connected, that 
they vibrate through equal angles, but in contrary directions ; 
by which means, the one must always be accelerated as much 
as the other is retarded by any external agitation. But as Mr. 
Cummins observes, when balances are connected by means of 
teeth, there arises a resistance which, however small, when ap- 
plied in this most delicate part, will tend to diminish the mo- 
mentum of the balances. 

40. That escapement is best in which the duration of the 
action of the balance-wheel on the pallets is least with respect 
to the time of vibration. 

Hence the detached escapement is the best, which appears 



DYNAMICS : ESCAPEMENTS. 277 

to have been the invention of the ingenious artist, Mr. Thomas 
Mudge, who made a watch on this construction for the late 
King of Spain, Ferdinand VI., in the year 1755. 

41. The time of the vibration of the balance is increased by 
heat, and diminished by cold. 

First, because the length of the spiral spring is increased by 
heat, and therefore its force diminished, and the contrary by 
cold. 2dly. The diameter of the balance is increased by heat, 
and therefore also the time of vibration ; and the contrary by 
cold. 

42. That balance is the most perfect which, without the com- 
pensation of a thermometer, is most subject to the influence of 
heat and cold. 

Because the obstructions from oil and friction act as a com- 
pensation to the expansion or contraction of the spring and 
balance ; therefore that balance which is most affected, is freest 
from the influence of oil and friction. 

43. The errors in the going of a watch, arising from the 
change of temperature, may be corrected by varying the length 
of the balance spring. 

Nevertheless, as it is extremely difficult to form an isochro- 
nal spiral, any variation in its length is dangerous, because we 
shall thus probably lose that point which determines its iso- 
chronism. 

44. The errors in the going of a watch, occasioned by the va- 
riation of temperature, may be corrected by varying the diame- 
ter of the balance. 

This may be effected by dividing the 
rim of the balance into two or more 
separate parts, g d, i f, h e, each of 
which is composed of two plates of me- 
tal of different expansibility, riveted 
together, the least expansible being near- 
est the centre n, and carrying at one end 
d, f, e, a weight ; whilst the other is con- 
nected either with the rim of the balance, 
or one of its radii. Now if the tempe- 
rature increase, the exterior plate expanding more than the in- 
terior, the compound will become more concave towards the 
centre ; and consequently the end which carries the weight will 
approach the centre of the balance, and on that account the 
vibrations will be rendered quicker. At the root of each ther- 
mometer, there is a screw g, i, h, by which the diameter of the 
balance may be increased or diminished, so as to alter the time 
kept by the chronometer, without interfering with the adjust- 
ment for heat and cold ; and if the magnitude and position of 
37 2 B 




278 DYNAMICS I SELECT MECHANICAL EXPEDIENTS. 

the weights be properly regulated, they will correct the error 
arising from the variation of the diameter of the balance caused 
by the variation of temperature. (M. Young's Analysis.) 
The reader who wishes to acquire practical knowledge on this 
subject, may advantageously consult Hat ton's Introduction to 
the Mechanical part of Clock and Watch Work. 



Select Mechanical Expedients. 

Although a full account of the principal contrivances for 
transmitting motion, and changing its rate, its direction, or its 
character, would carry us much beyond the assigned limits ; yet 
it seems advisable to give a few of these, which are, therefore, 
here presented. 

1. Spiral Gear. In the ordinary cases, the teeth of wheels 
are cut across their circumferences in a direction parallel to the 
axis. But, in the spiral gear, now used a little in this country, 
and still more in the American states, (especially in cotton- 
mills,) the teeth are cut obliquely, so that if 

they were continued they would pass round 
the axis like the threads of a screw. By rea- 
son of this disposition, the teeth come in con- 
tact only in the line of the centres, and thus 
operate, in great measure, without friction. 
It must, however, be remarked, that the action 
of these wheels is compounded of two forces, 
one of which acts in the direction of the plane 
of the wheel, the other in the direction of its 
axis. 

This spiral gearing is sometimes applied to clock-work, and 
has this peculiarity, that it admits of a smaller pinion than any 
other gearing. 

2. Change of rotatory velocity. It is sometimes necessary 
that a machine should be propelled with a velocity which is 
not equable, but continually changes in a given ratio. Thus, 
in cotton-mills, it is necessary that the speed of certain parts 
of the machinery should continually decrease from the begin- 
ning to the end of an operation. To accomplish this, two 
conical drums of equal size are placed with their axes parallel 







DYNAMICS I SELECT MECHANICAL EXPEDIENTS. 279 

to each other, and with their larger diameters in opposite direc- 
tions. They are connected by a belt, which 
is so regulated by proper mechanism, that n &n 

it is gradually moved from one extremity of r\ \ 7 

the conic frustums or drums to the other ; / \ \ j 
and thus acting upon circles of different 
diameter, causes a continual change of velo- 
city in the driven cone with relation to that 
which drives it. 

Thus, if the drum on the axis a b drives the wheel on the 
axis a b, and the belt commences its operation at the ends a a ; 
the driven conic frustum will first revolve slower than that 
which drives it, and will continue to move slower until the belt 
has reached the middle of both, when the rotatory motions of 
both will be equal : after that, the cone which is driven will 
turn quickest, and will so continue, turning quicker and quick- 
er both with respect to the other, and, in fact, until the belt 
reaches the ends b, b. 

A change of rotatory velocity, upon the same general prin- 
ciple, is sometimes effected thus. A decreasing series of toothed 
wheels is placed in the order of their size upon a common axis, 
bind fixed upon it. A corresponding series, but in an inverted 
order, are placed upon another axis, and not fixed, but capable 
of revolving about the axis like loose pulleys. The axis of 
this second series is hollow, and contains a moveable rod, 
which has a tooth projecting through a longitudinal slit in one 
side of the axis ; and this tooth serves to lock any one of the 
wheels by entering a notch cut for its reception. 
Thus, however, only one wheel can be locked 
at a time, the others remaining loose. Hence, 
the driven axis will revolve with a velocity 
which is due to the relative size of the wheel 
which is locked and that which drives it. Sup- 
pose, for example, that the diameters of the 
wheels are as 1, 2, 3, 4, and 5, upon the driving 
axis, and the corresponding wheels, upon the 
driven axis, are as the numbers 5,4, 3, 2, 1. Then, when the 
wheel 1 drives the wheel I, the rotatory velocity of the latter 
will be one-fifth of that of the former. When 2 drives II, the 
rotatory velocity communicated will be half that of the wheel 
2. When 3 drives III, the velocities will be equal. When 4 
drives IV, the rotatory velocity of the latter will be double that 
of the former. And, when 5 drives V, the velocity of the lat- 
ter will be five times that of the former. Different proportions 
in the diameters of the wheels will, of course, give different 
proportions in the velocities. 




280 



DYNAMICS : SELECT MECHANICAL EXPEDIENTS, 




We owe this beautiful contrivance to the late Mr. Bramah. 

It is sometimes requisite that a wheel or axis should move 
with different velocities in different 
parts of one and the same revolution. 
This may be accomplished by an ec- 
centric crown wheel acting upon and 
driving a long pinion. Thus, if the 
crown wheel in the margin rotates 
uniformly upon a centre of motion c, 
which is not the centre of the wheel, and the teeth of this crown 
wheel play into the leaves of the long pinion p q, since the por- 
tions of the crown wheel pass in contact with the pinion with 
different velocities, as their distances from the centre of motion 
c vary, the pinion p q will turn with an unequable or varying 
velocity, depending upon the eccentricity of the centre of mo- 
tion c. 

3. Cams and Wipers. These are contrivances by means of 
which beams placed vertically, or inclined aslant upwards, 
may be made to advance over a small space in the direction 
of their length, and then recede in the opposite direction ; 
and so on alternately. Eccentric circles, hearts, ellipses, 
portions of circles, and projecting epicycloids, serve to 
communicate these kinds of motions. Thus, in the first of 
the figures below, the circular eccentric cam, being put into 
uniform rotation, the sliding or reciprocating part a b of the 
machinery, will ascend and descend with a gentle, smooth 
motion ; being never at rest, unless at the very moment of 
changing its direction. In the quadrant cam, represented in 
the second figure, the reciprocating part a' b' will remain at 
rest on the periphery of the cam during the first quarter of 
the revolution, while, during the second, it will descend to 




the axis of motion ; during the third it will be at rest upon 
the axis ; and during the fourth it will return to its original 
situation. The elliptical cam, in the third figure, turning upon 



DYNAMICS : SELECT MECHANICAL EXPEDIENTS. 



281 



its centre, causes two alternate movements of a" b" for each re- 
volution of the ellipse. In the fourth figure, a triple cam is ap- 
plied to a tilt or trip hammer, turning upon a centre : there are 
three epicycloidal cams, or wipers, as in this case they are often 
called, causing three strokes of the hammer for one revolution 
of the wheel, to whose circumference these wipers are attached 
at equal distances. 

4. Parallel motions is a term given to the contrivances by 
which, especially in steam engines, circular motion, whether 
continued or alternate, is converted into alternate rectilinear 
motion, and vice versa. A moveable parallelogram is often, 
and very successfully, employed for this purpose ; as will be 
described when we speak of the steam engine. From among 
the numerous other contrivances for this purpose, we shall 
select only one, which is very simple and elegant ; and may 
be used in saw mills, and other 
reciprocating machines, as 
well as in steam engines. This 
is the invention of Dr. Cart- 
wright. The reciprocating 
motion of the piston rod or 
other rod m n, in the same 
rectilinear course is insured 
by connecting it with two 
equal cranks arranged in op- 
position to each other, and 
having their axes geared to- 
gether by two equal teethed 
wheels, w w, which play re- 
gularly into each other. 

5. Epicycloidal Wheel. — This is another very beautiful 
method of converting circular into alternate motion, or the 
contrary, a b is a fixed wheel, 
having teeth disposed uni- 
formly on its inner rim. c is 
a toothed wheel of half the 
diameter of the fixed wheel, its 
centre c revolving about the 
centre of the said fixed wheel. 
While this revolution of the 
wheel c is going on, any point 
whatever on its circumference 
will describe a straight line ; 
or will pass and repass through 

a diameter of the moving circle once during each revolution. 
This is an elegant application of the well known mathematical 

2b2 





282 



DYNAMICS : SELECT MECHANICAL EXPEDIENTS. 




property, that if a circle rolls on the inside of another of 
twice its diameter, the epicycloid described is a right line. 
In practice, the piston rod, or other reciprocating part, 
may be attached to any point on the circumference of the 
wheel c. 

6. Double Back and Pinion. — This is a contrivance for an 
alternating motion with a gradual change, a b is a double 
rack, with circular ends, fixed to a beam that is capable of 
moving in the direction of its 
length. The rack is driven 
by a pinion p, which is sus- 
ceptible of moving up and 
down in a groove mm', cut 
in the cross piece. When the 
pinion has moved the rack 
and beam until the end b is 

reached, the projecting a meets the spring s, and the rack is 
pressed against the pinion. Then the pinion, working in the 
circular end of the rack, will be forced down the groove 
m m' until it works in the lower side of the rack, and moves 
the beam back in the opposite direction ; and thus the motion 
is continued. The motion of the pinion in the groove will 
be diminished, if, instead of a double rack, there be used a 
single row of pins which are parallel to the axis of the pinion : 
this plan is sometimes adopted in the machines called man- 
gles. 

7. The Universal Lever. — This is a 
French invention, and is often, from 
the name of the inventor, called lever 
de la Garousse. It consists of a bar 
a b, moving upon a centre c, and 




having a moveable catch, or hook, h h' 
attached to each side, and acting upon 
the oblique teeth of a double rack ; so 
that, as b and a alternately rise and fall in 
the reciprocal motion, the hooks h and h' 
successively lay hold of the teeth of the 
beam a b, and draw it up in the direction b a. 

8. The Tachometer. — This is a very ingenious contrivance, 
which we owe to Mr. Donkin, for the purpose of measuring 
small variations in velocity, The contrivance is, in fact, hydro- 
dynamical, but we mention it here, and the simplicity of its 
principle will render it easy of comprehension. 

If a cup with any fluid, as mercury, be placed upon a 
spindle, so that the brim of the cup shall revolve horizon- 
tally round its centre, then the mercury in the cup will 



DYNAMICS : SELECT MECHANICAL EXPEDIENTS. 283 

assume a concave form, that is, the mercury will rise on the 
sides of the cup, and be depressed in the middle ; and the more 
rapid the rotation of the cup, the more will the surface of the 
mercury be depressed in the middle and rise at the sides ; the 
figures being those of hollow paraboloids. Now, if the mouth of 
this cup be closed, and a tube inserted into it, terminated in the 
cup by a ball-shaped end, and half filled with some coloured 
liquid, as coloured spirits of wine ; then it is evident that the 
more the surface of the mercury is depressed the more the fluid 
in the tube will fall, and vice versa. Consequently, the velocity 
of rotation of the cup, and of the spindle to which it is attached, 
will be indicated by the height of the liquid in the tube ; and, 
indeed, absolutely measured by it, when the apparatus has been 
subjected to the adequate preparatory experiments. 

For a more minute description, see Trans. Soc. of Arts, vol. 
xxvi., or the article Tachometer in the Pantologia. 

Two plates, exhibiting a great variety of contrivances for 
converting rotatory, reciprocating, and other motions, one into 
the other, and thus facilitating the construction of machinery, 
are given in my Treatise of Mechanics. 

A large and valuable plate of the same kind, exhibiting 178 
useful elementary mechanical combinations, has been lately pub- 
lished at Manchester, and is sold in London by Ackerman. 
I scarcely know a more interesting present than this would be 
to a young mechanic. 



284 HYDROSTATICS. 

CHAPTER XI, 
HYDROSTATICS. 

1. Hydrostatics comprises the doctrine of the pressure and 
the equilibrium of non-elastic fluids, as water, mercury, &c. 
and that of the weight and pressure of solids immersed in 
them. 

2. Def. A fluid is a body whose parts are very minute, 
yield to any force impressed upon it (however small), and by 
so yielding are easily moved among themselves. 

Some attempt to give mechanical ideas of a fluid body by 
comparing it to a heap of sand : but the impossibility of giving 
fluidity by any kind of mechanical comminution will appear by 
considering two of the circumstances necessary to constitute a 
fluid body : 1. That the parts, notwithstanding any compres- 
sion, may be moved in relation to each other, with the smallest 
conceivable force, or will give no sensible resistance to motion 
within the mass in any direction. 2. That the parts shall gra- 
vitate to each other, whereby there is a constant tendency to 
arrange themselves about a common centre, and form a spherical 
body ; which, as the parts do not resist motion, is easily exe- 
cuted in small bodies. Hence the appearance of drops always 
takes place when a fluid is in proper circumstances. It is ob- 
vious that a body of sand can by no means conform to these 
circumstances. 

Different fluids have different degrees of fluidity, according 
to the facility with which the particles may be moved amongst 
each other. Water and mercury are classed among the most 
perfect fluids. Many fluids have a very sensible degree of tena- 
city, and are therefore called viscous or imperfect fluids. 

3. Def. Fluids may be divided into compressible and in- 
compressible, or elastic and non-elastic fluids. A compressi- 
ble or elastic fluid is one whose apparent magnitude is dimi- 
nished as the pressure upon it is increased, and increased by a 
diminution of pressure. Such is air, and the different vapours. 
An incompressible or non-elastic fluid is one whose dimensions 
are not, at least as to sense, affected by any augmentation of 
pressure. Water, mercury, wine, &c. are generally ranged un- 
der this class. 

It has been of late years proposed to limit the application 
of the term fluids to those which are elastic, and to apply the 



HYDROSTATICS : PRESSURE OF FLUIDS. 285 

word liquid to such as are non-elastic. But it is an unnecessary- 
refinement. 

4. Def. The specific gravity of any solid or fluid body is 
the absolute weight of a known volume of that substance, 
namely, of that which we take for unity in measuring the capa- 
cities of bodies. 

Comparing this definition with that of density (Dynamics, 
Def. 2), it will appear that the two terms express the same thing 
under different aspects. 



Section I. — Pressure of Non-elastic Fluids. 

1. Fluids press equally in all directions, upwards, downwards, 
aslant, or laterally. 

This constitutes one essential difference between fluids and 
solids, solids pressing only downwards, or in the direction of 
gravity. 

2. The upper surface of a gravitating fluid at rest is hori- 
zontal. 

3. The pressure of a fluid on every particle of the vessel con- 
taining it, or of any other surface, real or imaginary, in contact 
with it, is equal to the weight of a column of the fluid, whose 
base is equal to that particle, and whose height is equal to its 
depth below the upper surface of the fluid. 

4. If, therefore, any portion of the upper part of a fluid be 
replaced by a part of the vessel, the pressure against this from 
below will be the same which before supported the weight of 
the fluid removed, and every part remaining in equilibrium, the 
pressure on the bottom will be the same as it would if the ves- 
sel were a prism or a cylinder. 

5. Hence, the smallest given quantity of a fluid may be made 
to produce a pressure capable of sustaining any proposed weight, 
either by diminishing the diameter of the column and increas- 
ing its height, or by increasing the surface which supports the 
weight. 

6. The pressure of a fluid on any surface, whether vertical, 
oblique, or horizontal, is equal to the weight of a column of the 
fluid whose base is equal to the surface pressed, and height 
equal to the distance of the centre of gravity of that surface be- 
low the upper horizontal surface of the fluid. 

7. Fluids of different specific gravities that do not mix, will 
counterbalance each other in a bent tube, when their heights 
above the surface of i unction are inversely as their specific 
gravities. 

A portion of fluid will be quiescent in a bent tube, when 
38 



286 HYDROSTATICS I PRESSURE OF FLUIDS. 

the upper surface in both branches of the tube is in the same 
horizontal plane, or is equidistant from the earth's centre. And 
water poured down one branch of such a tube (whether it be of 
uniform bore throughout, or not) will rise to its own level in the 
other branch. 

Thus water may be conveyed by pipes from a spring on 
the side of a hill, to a reservoir of equal height on another 
hill. 

8. The ascent of a body in a fluid of greater specific gravity 
than itself, arises from the pressure of the fluid upwards against 
the under surface of the body. 

9. Def. The centre of pressure is that point of a surface 
against which any fluid presses, to which if a force equal to the 
whole pressure were applied it would keep the surface at rest, 
or balance its tendency to turn or move in any direction. 

10. If a plane surface which is pressed by a fluid be pro- 
duced to the horizontal surface of it, and their common in- 
tersection be regarded as the axis of suspension, the centres 
of percussion and of pressure will be at the same distance 
from the axis. 

11. The centre of pressure of a parallelogram, whose upper 
side is in the plane of the horizontal level of the liquid, is at § 
of the line (measuring downwards) that joins the middles of the 
two horizontal sides of the parallelogram. 

12. If the base of a triangular plane coincides with the 
upper surface of the water, then the centre of pressure is at the 
middle of the line drawn from the middle of the base to the 
vertex of the triangle. But, if the vertex of the triangle be in 
the upper surface of the water, while its base is horizontal, the 
centre of pressure is at § of the line drawn from the vertex 
to bisect the base. 



Illustrations and Applications. 



1. If several glass tubes of different shapes 
and sizes be put into a larger glass vessel 
containing water, the tubes being all open at 
top ; then the water will be seen to rise to the 
same height in each of them, as is marked 
by the upper surface a c, of the liquid in the 
larger vessel. 

2. If three vessels of equal bases, one cylindrical, the 




HYDROSTATICS : PRESSURE OF FLUIDS. 287 

second considerably larger at top than at bottom, the third con- 
siderably less at top than at bottom, and with the sides of the 
two latter either regularly or irregularly sloped, have their 
bottoms moveable, but kept close by the action of a weight upon 
a lever ; then it will be found, that when the same weight acts 
at the same distance upon the lever, water must be poured in to 
the same height in each vessel before its pressure will force 
open the bottom. 

3. Let a glass tube, open at both ends (whether cylindrical 
or not, does not signify), have a piece of bladder tied over one 
end, so as to be capable of hanging below that end, or of rising 
up within it, when pressed from the outside. Pour into this 
tube some water tinged red, so as to stand at the depth of seven 
or eight inches ; and then immerse the tube with its coloured 
water vertically into a larger glass vessel nearly full of colour- 
less water, the bladder being downwards serving as a flexible 
bottom to the tube. Then, it will be observed that when the 
depth of the water in the tube exceeds that in the larger vessel, 
the bladder will be forced below the tube, by the excess of the 
interior over the exterior pressure : but when the exterior water 
is deeper than the interior, the bladder will be thrust up within 
the tube, by the excess of exterior pressure : and when the water 
in the tube and that in the larger vessel, have their upper sur- 
faces in the same horizontal plane, then the bladder will adjust 
itself into a flat position just at the bottom of the tube. The 
success of this experiment does not depend upon the actual 
depth of the water in the tube, but upon the relation between 
the depths of that and the exterior water ; and proves that in all 
cases, the deeper water has the greater pressure at its bottom, 
tending equally upward and downward. 

4. The hydrostatical paradox, as it is usually denominated, 
results from the principle that any quantity of a non-elastic 
fluid, however small, may be made to balance another quan- 
tity ; or any weight, as large as we please. It may be illus- 
trated by the hydrostatic bellows, consisting of two thick boards 
d c, f e, each about 16 or 18 inches dia- 
meter, more or less, covered or connected 
firmly with leather round the edges, to 
open and shut like a common bellows, but 
without valves ; only a pipe a b, about 3 
feet high, is fixed into the bellows above 
f. Now let water be poured into the pipe 
at a, and it will run into the bellows, gra- 
dually separating the boards by raising 
the upper one. Then if several weights, 
as three hundred weights, be laid upon the upper board, by 




288 HYDROSTATICS I PRESSURE OP FLUIDS. 

pouring the water in at the pipe till it be full, it will sustain all 
the weights, though the water in the pipe should not weigh a 
quarter of a pound : for the pipe or tube may be as small as we 
please, provided it be but long enough, the whole effect depend- 
ing upon the height, and not at all on the width of the pipe : for 
the proportion is always this, 

As the area of the orifice of the pipe 

is to the area of the bellows board, 

so is the weight of water in the pipe, above d c, 

to the weight it will sustain on the board. 

5. In lieu of the bellows part of the apparatus, the leather of 
which would be incapable of resisting any very considerable 
pressure, the late Mr. Joseph Bramah used a very strong me- 
tal cylinder, in which a piston moved in a perfectly air and 
water tight manner, by passing through leather collars, and as 
a substitute for the high column of water, he adopted a very 
small forcing pump to which any power can be applied ; and 
thus the pressing column becomes indefinitely long, although 
the whole apparatus is very compact, and takes but little room. 
The marginal figure is a section of one 
of these presses, in which c is the piston 
of the large cylinder, formed of a solid 
piece of metal turned truly cylindrical, 
and carrying the lower board v of the 
press upon it : u is the piston of the small 
forcing pump, being also a cylinder of 
solid metal moved up and down by the 
handle or lever iv. The whole lower 
part of the press is sometimes made to 
stand in a case x x, containing more 
than sufficient water as at y, to fill both the cylinders ; and 
the suction pipe of the forcing pump u dipping into this water 
will be constantly supplied. Whenever, therefore, the handle 
w is moved upwards, the water will rise through the conical 
metal valve z, opening upwards into the bottom of the pump 
u ; and when the handle is depressed, that water will be forced 
through another similar valve a, opening in an opposite di- 
rection in the pipe of communication between the pump and 
the great cylinder b, which will now receive the water by which 
the piston rod t will be elevated at each stroke of the pump u. 
Another small conical valve c is applied by means of a screw 
to an orifice in the lower part of the large cylinder, the use of 
which is to release the pressure whenever it may be necessary ; 
for, on opening this valve, any water which was previously 




HYDROSTATICS ! BRAMAh's PRESS. 289 

contained in the large cylinder b, will run off into the reservoir 
y by the passage d, and the piston t will descend ; so that the 
same water may be used over and over again. The power of 
such a machine is enormously great ; for, supposing the hand 
to be applied at the end of the handle w, with a force of only 
10 pounds, and that this handle or lever be so constructed as to 
multiply that force but 5 times, then the force with which the 
piston it descends will be equal to 50 pounds : let us next sup- 
pose that the magnitude of the piston t is such, that the area of 
its horizontal section shall contain a similar area of the smaller 
piston u 50 times, then 50 multiplied by 50 gives 2500 pounds, 
for the force with which the piston t and the presser v will 
rise. A man can, however, exert ten times this force for a 
short time, and could therefore raise 25,000 pounds ; and 
would do more if a greater disproportion existed between the 
two pistons t and u, and the lever w were made more favour- 
able to the exertion of his strength. 

This machine not only acts as a press, but is capable of many 
other useful applications, such as a jack for raising heavy loads, 
or even buildings ; to the purpose of drawing up trees by their 
roots, or the piles used in bridge-building. 

To find the thickness of the metal in Bramah's press, 
to resist certain pressures, Mr. Barlow gives this theorem, 

7) T 

t = —*- — - where p = pressure in lbs. per square inch, r = 

radius of the cylinder, t = its thickness, and c = 18000 lbs. 
the cohesive power of a square inch of cast iron. 

Ex. Suppose it were required to determine the thickness of 
metal in two presses, each of 6 inches radius, in one of which 
the pressure may extend to 42 7S pounds, in the other to 8556 
pounds per square inch. 

Here in the first case, 

4278 X 6 _ a „ . ' ... . 

t = = 1*87 inches, thickness. 

18000—4278 ' 

In the second, 

t — = 5-43 inches, thickness. 

18000 — 8556 

The usual rules, explained below (art. 10) would make the 
latter thickness double the former : extensive experiments are 
necessary to tell which method deserves the preference. 

6. If b the breadth, and d the depth of a rectangular gate, 
or other surface exposed to the pressure of water from top to 
bottom ; then the entire pressure is equal to the weight of a 
prism of water whose content is h b d 2 . Or, if b and d be in 

2C 



290 HYDROSTATICS : PRESSURE OP SLUICE-GATES, &C. 

feet, then the whole pressure = 31 J b d 2 , in lbs. or nearly = 
T 3 T b d 2 , in cwts. 

7. If the gate be in form of a trapezoid, widest at top, then, if 
b and b be the breadths at the top and bottom respectively, and 
d the depth. 

whole pressure in lbs. = 3H [i (b — b) -f b~\ d 2 

whole pressure in cwts. = T 3 T [§ (b — b) + b] d 2 , nearly. 

8. The weight of a cubic foot of rain or river water, is nearly 
equal to T 6 T cwt. 

The pressure on a square inch, at the depth of THiR-/y feet 
is very nearly Tmn-teen pounds. 

Pressure on a square foot, nearly a ton at the depth of thir- 
ty-six feet. [The true depth is 35*84 feet] 

The weight of an ale gallon of rain water is nearly \0\ lbs. 
that of an imperial gallon 10 lbs. 

The weight of a cubic foot of sea-water is nearly 4 of a cwt. 

These are all useful approximations. 

Thus, the pressure of rain water upon a square inch at the 
depth of 3000 feet, is 1300 lbs. 

And the pressure upon a square foot at the depth of 108 feet 
is nearly three tons. 

9. In the structure of dykes or embankments, both faces or 
slopes should be planes, and the ex- 
terior and interior slopes should make 
an angle of not less than 90°. For if 
a d' be the exterior slope, and the an- 
gle d' a b be acute, e d' perpendicular D f 
to a b is the direction of the pressure 
upon it \ and the portion d' a e will probably be torn off. But 
when d a is the exterior face, making with a b an obtuse angle, 
the direction of the pressure falls within the base, and there- 
fore augments its stability. 

10. The strength of a circular bason confining water, requires 
the consideration of other principles. 

The perpendicular pressure against the wall depends merely 
on the altitude of the fluid, without being affected by the 
volume. But, as professor Leslie remarks, the longitudinal 
effort of the thrust, or its tendency to open the joints of the 
masonry, is measured by the radius of the circle. To resist 
that action in very wide basons, the range or course of stones 
along the inside of the wall, must be proportionally thicker. 
On the other hand, if any opposing surface present some con- 
vexity to the pressure of water, the resulting longitudinal strain 
will be exerted in closing the joints and consolidating the build- 
ing. Such reversed incurvation is, therefore, often adopted in 
the construction of dams. 




HYDROSTATICS *. EMBANKMENTS, PIPES, &C. 291 

In like manner, the thickness of pipes to convey water 

must vary in proportion to — , where h is the height of the 

head of water, d the diameter of the pipe, and c the measure of 
the cohesion of a bar of the same material as the pipe, and an 
inch square. 

A pipe of cast iron, 15 inches diameter, and I of an inch 
thick, will be strong enough for a head of 600 feet. 

A pipe of oak of the same diameter, and 2 inches thick, 
would sustain a head of 180 feet. 

Where the cohesion is the same, t varies as h d : or as 
h d : t :: h d : t, in the comparison of two cases.* 

Example. — What, then, must be the respective thicknesses 
of pipes of cast iron and oak, each 10 inches diameter, to carry 
water from a head of 360 feet ? 

Here, 1 st. for cast iron : 

h d (= 600 X 15) : t (= |) :: h d ( = 360 X 10) : t = 
360 X 10 X 3 10800 , . ' 

= T^rrrz — -rs ■» an inch. 

600 X 15 X 4 36000 10 

2dly. for oak: 
h d (= 180 X 15) : t (= 2) :: h d ( = 360 X 10 ) : t = 

360 X 1Q X 2 - tf - f - 4 inches. 
180 X 15 15 3 3 



Section II. — Floating Bodies. 

1. If any body float on a fluid, it displaces a quantity of the 
fluid equal to itself in weight. 

2. Also, the centres of gravity of the body and of the fluid 
displaced must, when the body is at rest, be in the same verti- 
cal line. 

3. If a vessel contain two fluids that will not mix (as water 
and mercury), and a solid of some intermediate specific gravity 
be immersed under the surface of the lighter fluid and float on 
the heavier ; the part of the solid immersed in the heavier fluid, 
is to the whole solid as the difference between the specific gra- 

* To ascertain whether or not a pipe is strong enough to sustain a proposed 
pressure, it is a good custom amongst practical men to employ a safety valve, usually 
of an inch in diameter, and load it with the proposed weight, and a surplus deter- 
mined by practice. Then, if the proposed pressure be applied interiorly, by a forcing 
pump, or in any other way, if the pipe remain sound in all its parts after the safe- 
ty-valve has yielded, such pipe is regarded as sufficiently strong. 

The actual pressures upon a pipe of any proposed diameter and head, may evi- 
dently be determined by a similar method. 



292 



HYDROSTATICS I FLOATING BODIES. 



vities of the solid and the lighter fluid, is to the difference be- 
tween the specific gravities of the two fluids. 

4. The buoyancy of casks, or the load which they will carry 
without sinking, may be estimated by reckoning 10 lbs. avoir- 
dupois to the ale gallon, or 8± lbs. to the wine gallon. 

5. The buoyancy of pontoons may be estimated at about half 
a hundred weight for each cubic foot. 

Thus a pontoon which contained 96 cubic feet, would sustain 
a load of 48 cwt. before it would sink. 

N. B. This is an approximation, in which the difference be- 
tween T 6 T and h, that is, ^ ¥ of the whole weight, is allowed for 
that of the pontoon itself. 

6. The principles of buoyancy are very ingeniously applied 
in Mr. Farey's self-acting flood-gate. In the case of common 
sluices to a mill-dam, when a sudden flood occurs, unless the 
miller gets up in the night to open the gate or gates, the 
neighbouring lands may become inundated ; and, on the con- 
trary, unless he be present to shut up when the flood subsides, 
the mill-dam may be emptied and the water lost which he 
would need the next day. To prevent either of these occur- 
rences, Mr. John Farey, whose talent and ingenuity are well 
known, has proposed a self-acting flood-gate, the following 
description of which has been given in the Mechanics' Weekly 
Journal. 

a a represents a vertical section of a gate poised upon a 
horizontal axis passing rather above the centre of pressure of 
the gate, so as to give it a tendency to shut close, a a is a 




HYDROSTATICS I FAREY's FLOOD-GATE. 293 

lever, fixed perpendicular to the gate, and connected by an iron 
rod with a cask, b, floating upon the surface of the water, when 
it rises to the line, b, d, which is assumed as a level of the wear 
or mill-dam, b, c, e, f, in which the flood-gate is placed : by 
this arrangement it will be seen that when the water rises 
above the dam, it floats the cask, opens the gate, and allows the 
water to escape until its surface subsides to the proper level 
at b, d ; the cask now acts by its weight, when unsupported by 
the water, to close the gate and prevent leakage. The gate 
should be fitted into a frame of timber, h, k, which is set in the 
masonry of the dam. The upper beam, h, of the frame being 
just level with the crown of the dam, so that the water runs 
over the top of the gate at the same time that it passes through 
it : to prevent the current disturbing the cask, it is connected 
by a small rod, e, at each end, to the upper beam, h, of the frame, 
and jointed in such a manner as to admit of motion in a vertical 
direction. 

Any ingenious mechanic will so understand the construction 
from this brief account, as to be able to apply it to practice 
when needed. 

7. By means of the same principle of buoyancy it is, that 
a hollow ball of copper attached to a metallic lever of about a 
foot long, is made to rise with the liquid in a water-tub, and 
thus to close the cock and stop the supply from the pipe, just 
before the time when the water would otherwise run over the 
top of the vessel. 

8. This property, again, has been successfully employed in 
pulling up old piles in a river where the tide ebbs and flows. 
A barge of considerable dimensions is brought over a pile as 
the water begins to rise : a strong chain which has been pre- 
viously fixed to the pile by a ring, &c. is made to gird the 
barge, and is then fastened. As the tide rises the vessel rises too, 
and by means of its buoyant force draws up the pile with it. 

In an actual case, a barge 50 feet long, 12 feet wide, 6 deep, 

and drawing two feet of water, was employed. Here, 

50 X 12 X 16 
50 X 12 X (6 — 2) X 4 = - = 192 X 7{ = 

1344 4- 27f = 1371-f cwt. = 66% tons nearly, the measure of 
the force with which the barge acted upon the pile. 



Section III. — Specific Gravities. 

1. If a body float on a fluid, the part immersed is to the whole 
body, as the specific gravity of the body to the specific gravity 
of the fluid. 

39 2 c 2 



294 HYDROSTATICS : SPECIFIC GRAVITIES. 

Hence, if the body be a square or a triangular prism, and it 
be laid upon the fluid, the ratio of that portion of one end which 
is immersed, to the whole surface of that end, will serve to de- 
termine the specific gravity of the body. 

2. If the same body float upon two fluids in succession, the 
parts immersed will be inversely as the specific gravities of 
those fluids. 

3. The weight which a body loses when wholly immersed 
in a fluid is equal to the weight of an equal bulk of the 
fluid. 

When we say that a body loses part of its weight in a fluid, 
we do not mean that its absolute weight is less than it was 
before, but that it is partly supported by the reaction of the 
fluid under it, so that it requires a less power to sustain or to 
balance it. 

4. A body immersed in a fluid ascends or descends with a 
force equal to the difference between its own weight and the 
weight of an equal bulk of fluid ; the resistance or viscosity of 
the fluid not being considered. 

5. To find the specific gravity of a fluid or of a solid. — 
On one arm of a balance suspend a globe of lead by a fine thread, 
and to the other fasten an equal weight, which may just balance 
it in the open air. Immerse the globe into the fluid, and ob- 
serve what weight balances it then, and consequently what 
weight is lost, which is proportional to the specific gravity as 
above. And thus the proportion of the specific gravity of one 
fluid to another is determined by immersing the globe succes- 
sively in all the fluids, and observing the weights lost in each, 
which will be the proportions of the specific gravities of the 
fluids sought. 

This same operation determines also the specific gravity of 
the solid immerged, whether it be a globe or of any other shape 
or bulk, supposing that of the fluid known. For the specific 
gravity of the fluid is to that of the solid, as the weight lost is 
to the whole weight. 

Hence also may be found the specific gravity of a body that 
is lighter than the fluid, as follows : 

6. To find the specific gravity of a solid that is lighter 
than the fluid, as water, in which it is put. — Annex to the 
lighter body another that is much heavier than the fluid, so as 
the compound mass may sink in the fluid. Weigh the heavier 
body and the compound mass separately, both in water and out 
of it ; then find how much each loses in water, by subtracting 
its weight in water from its weight in air ; and subtract the less 
of these remainders from the greater. 



HYDROSTATICS I SPECIFIC GRAVITIES. 295 

Then, as this last remainder, 

Is to the weight of the light body in air, 
So is the specific gravity of the fluid, 
To the specific gravity of that body. 

7. The specific gravities of bodies of equal weight, are reci- 
procally proportional to the quantities of weight lost in the same 
fluid. And hence is found the ratio of the specific gravities 
of solids by weighing in the same fluids masses of them that 
weigh equally in air, and noting the weights lost by each. 

8. Instead of a hydrostatic balance, a hydrostatic steelyard 
is now frequently employed. It is contrived to balance ex- 
actly by making the shorter end wider, and with an enlarge- 
ment at the extremity. The shorter arm is undivided, but the 
longer arm is divided into short equal divisions : thus, if that 
longer arm be 8 inches long, it may be divided into 400 parts, 
the divisions commencing at a. Then, in using this instru- 




ment, any convenient weight is suspended by a hook from a 
notch at the end of the scale a. The body whose specific gra- 
vity is to be determined, is suspended from the other arm by a 
horse-hair, and slid to and fro till an equilibrium is produced. 
Then, without altering its situation at d in the beam, it is im- 
mersed in water, and balanced a second time by sliding the 
counterpoise from a, say to c. 

Here, evidently, weight in water : weight in air :: b c : b a : 
and loss of weight in water : weight in air : : a c : a b. 

- weight in air a b .„ 

Conseq. 2= == = specific gravity. 

loss AC 

With such an instrument nicely balanced upon a convenient 
pedestal, I find that the specific gravities of solids are ascertain- 
able both with greater facility and correctness than with any 
hydrostatic balance which I have seen.* 

* We owe this contrivance to Dr. Coates, of Philadelphia. 



296 



HYDROSTATICS I SPECIFIC GRAVITIES. 



Table of Specific Gravities. 











Weight cub 


MET/ 




Spec. 


inch, in 








Grav. 


avoird. oz. 


Arsenic 


- 


- 


5,763 


3-335 


Cast antimony- 


- 


- 


6,702 


3-878 


Cast zinc 


- 


- 


7,190 


4-161 


Cast iron 


- 


- 


7,207 


- - - 4-165 


Cast tin 


- 


- 


7,291 


4-219 


Bar iron 


- 


. 


7,788 


4-507 


Cast nickel - 


m 


- 


7,807 


4-513 


Cast cobalt - 


- 


- 


7,811 


4-520 


Hard steel 


- 


- 


7,816 


4-523 


Soft steel 


- 


- 


7,833 


4-533 


Cast brass 


. 


- 


8,395 


4-858 


Cast copper - 


- 


- 


8,788 


5-085 


Cast bismuth 


- 


- 


9,822 


5-684 


Pure cast silver 


- 


- 


- 10,474 


6-061 


Same hammered 


- 


- 


- 10,510 


6-082 


Cast lead 


- 


- 


11,352 


6-569 


Mercury 


- 


- 


13,568 


7-872 


Trinket gold 


- 


- 


- 15,709 


9-901 


Gold coin 


- 


- 


- 17,647 


- - 10.212 


Pure cast gold 


- 


- 


- 19,258 


- - 11-145 


Same hammered 


- 


- 


- 19,361 


- - 11-212 


Pure platinum 


- 


- 


19,500 


- - 11-285 


Same hammered 


- 


- 


- 20,336 


- - 11-777 


Platinum wire 


- 


. 


- 21,041 


- - 12176 


Platinum laminated, > 
or beat into leaves 3 


- 


- 22,069 


- - 12-763 


11 metals become more dense 


or heavy by hammering. 


STONES, EARTHS, &c 




Spec. 


Weight cub. 








Grav. 


foot avoird. lbs. 


Ambergris - 


- 


- 


926 




Amber 


- 


- 


1,078 




Phosphorous 


- 


- 


1,714 




Brick - 


- 


- 


2,000 - 


- - 12500 


Sulphur 


- 


- 


2,033 - 


- - 127-06 


Opal - 


- 


- 


2,114 





hydrostatics: specific gravities. 



297 



STONES, &c. continued. 




Spec. 


Weight cub. 








Grav. 


foot avoird. lb. 


Gypsum, opaque 


- 


- 


2,168 - 


- - 135-50 


Stone, paving 


- 


- 


2,416 - 


- - 151-00 


Mill-stone - 


- 


- 


2,484 - 


- - 155-20 


Stone, common - 


- 


- 


2,520 - 


- - 157-50 


Flint and spar 


- 


- 


2,594 - 


- - 162-12 


Crystal 


- 


- 


2,653 




Granite, red Egyptian 


- 


- 


2,654 - 


- - 165-87 


Glass, green 


- 


- 


2,642 




white 


- 


- 


2,892 




bottle 


- 


- 


2,733 




Pebble 


- 


- 


2,664 - 


- - 166*50 


Slate - 


- 


- 


2,672 - 


- - 167-00 


Pearl 


- 


- 


2,684 




Alabaster - 


- 


- 


2,730 




Marble 


- 


- 


2,742 - 


- - 171-38 


Porphyry - 


- 


- 


2,765 - 


- - 172-81 


Emerald 


- 


- 


2,775 




Chrysolite, jewellers' 


- 


- 


2,782 




Chalk 


- 


- 


2,784 - 


- - 174-00 


Jasper 


- 


- 


2,816 




Basaltes (Giant's causey) 


- 


2,864 - 


- - 17900 


Hone, white razor 


- 


- 


2,876 - 


- - 179-75 


Limestone 


- 


- 


3,179 - 


- - 198-68 


Flint glass 


. 


- 


3,329 




Diamond 


- 


- 


3,521 




Beryl 


- 


- 


3,549 




Sapphire 


- 


- 


3,994 




Topaz 


- 


- 


4,011 




Garnet 


- 


- 


4,189 




Ruby 






4,283 





RESINS, GUMS, &c. 

Grav. 

Wax 897 

Tallow 945 

Bees' wax 965 

Camphor 989 

Honey - - - - - - - 1456 

Bone of an ox ------ 1659 

Ivory 1822 



298 HYDROSTATICS I SPECIFIC GRAVITIES. 

LIQUIDS. Spec. 

Grav. 

Air, at the earth's surface, about - If 

Sulphuric Ether - - - - - 716 

Alcohol absolute - - - - - 792 

Liquid Bitumen ----- 848 

Oil of Turpentine 870 

Muriatic Ether 874 

Olive Oil ---.-- 915 

Burgundy Wine - - - - 991 

Wine of Bourdeaux - - - - - 994 

Port 997 

Distilled Water 1,000 

Sea Water 1,028 

Milk 1,030 

Beer 1,034 

Madeira Wine 1,038 

Nitric Acid 1,218 

Water from the Dead Sea - - - - 1,240 

Nitrous Acid ------ 1,550 

Sulphuric Acid 1,841 



WOODS. 


Spec. 


Weight cub. 




Grav. 


foot avoird. lb. 


Cork - - - - 


240 


15-00 


Poplar - - - - 


383 


23-94 


Larch - 


544 


34-00 


Elm, and N. E. Fir 


556 


34-75 


Mahogany, Honduras 


560 


35-00 


Poon - 


579 


36-18 


Willow - - - - 


585 


36-56 


Cedar - - - - 


596 


37-25 


Pitch Pine 


660 


41-25 


Pear tree - - - 


661 


41-31 


Walnut 


671 


- - - 41-94 


Mar Forest Fir 


694 


43-37 


Elder tree - - - 


695 


43-44 


Beech - - - - 


696 


43-50 


Orange wood - - - 


705 


- - - 44-06 


Cherry tree - - - 


715 


44-68 


Teak - - - - 


745 


- - - 46-56 


Maple and Riga Fir 


750 


- - - 46-87 


Ash and Dan. Oak 


760 


- - - 47*50 



HYDROSTATICS I SPECIFIC GRAVITIES. 299 

WOODS continued. Spec Weight cub. 

Grav. foot avoird. lb. 

Yew, Dutch - - - - 788 49-25 

Apple tree - - - - 793 49*56 

Alder 800 50*00 

Yew, Spanish - - - - 807 50'44 

Mahogany, Spanish - - 852 53*25 

Oak, Canadian - - - 872 54*50 

Box, French - - - - 912 57*00 

Logwood + 913 57*06 

Oak, English - - - * 970 60*62 

Ditto, 60 years old - 1170 73*12 

Ebony - 1331 83*18 

Lignum Vitse - 1333 83*31 

9. Since a cubic foot of water, at the temperature of 40° Fah- 
renheit, weighs 1000 ounces avoirdupois, or 62£ lbs., the num- 
bers in the preceding tables under the head Spec. Grav. exhibit 
very nearly the respective weights in avoirdupois ounces of a 
cubic foot of the several substances. 

We have also given, in another column, the weight in ounces 
of a cubic inch of each of the several metals : and, with re- 
gard to different kinds of wood and stone, the weight in avoir- 
dupois pounds of a cubic foot of each. These, of course, are 
medium specific gravities and weights ; for there are variations, 
sometimes indeed considerable ones, between the specific gra- 
vities of different specimens of the same kind of substance. 

These additional columns will evidently facilitate the labour 
of finding the magnitude of a body from its weight, or the 
weight of a body from its magnitude. In this respect, too, 
the following particulars will often be of utility. 

10. (1) 430*25 cubic inches of cast iron weigh 1 cwt. 

397*60 bar iron 

368*88 cast brass 

352*41 cast copper 

272*8 cast lead 

(2) 14*835 cubic feet of paving stone weigh a ton 

14*222 common stone 

13*505 granite 

13*070 marble 

12*874 chalk 

11*273 limestone 

64*460 elm 

64*000 . Honduras mahogany 

51*650 Mar Forest fir 

51*494 beech 



300 HYDROSTATICS : SPECIFIC GRAVITIES. 

47*762 cubic feet of Riga fir 

47*158 ash, and Dantzic oak 

42*066 Spanish mahogany 

36*205 English oak. 

11. Prob. To find the internal diameter of a uniform capil- 
lary or other small tube. 

Let the tube be weighed when empty, and again when filled 
with mercury, and let w be the difference of those weights in 
troy grains, and / the length of the tube in inches. Then the 

I to 
diameter required, c?= -019252 I—. 

Thus, if the difference of weights were 500 grains, and 
the length of the tube were 20 inches : we should have d= 

•019252 I — = -019252 X5=*09626 of an inch.* 



J 500 _ 
~20~~ 



12. Prob. To find the weight of a leaden pipe. 

If / be the length in feet, d the interior diameter, and t the 
thickness both in inches and parts of an inch, w the weight in 
hundred weights : then w = *1382 It (d+t). 

For a cast iron pipe, the theorem is 

w=*0876 It {d+t) ; 
or nearly T 7 T of the former expression. 

Ex. Let the internal diameter of a leaden pipe be 4 inches, 
the thickness \ of an inch ; required the weight of 12 feet in 
length. 

Here *1382 X 12 Xix4^ = *1382x 121 = 1*762 cwts. 

13. Prob. To find the weight of the ring or rim of a cast iron 
fly-wheel. 

Supposing this ring or annulus to be similar to a portion of a 
pipe of large diameter, the expression for the weight will be 
similar to the above ; but it may be advantageous to change 
the notation. Let then d be the interior diameter of the fly 
in inches, d half the difference of the exterior and interior 
diameters, t the thickness from side to side of the fly, and w its 
weight in hundred weights : then w=*0073 t d (v + d). 

* The same thing may easily be accomplished thus : — Let a cone of box wood, or 
of brass, be very accurately turned, of about 6 inches in length, and the diameter 
of its base about a quarter of an inch ; and let its curve surface be very accurately 
marked with a series of parallel rings, about a twentieth of an inch asunder, from 
its vertex to its base. Insert this cone 
carefully in the cylinder (so that then- 
axes shall coincide) as in the diagram : A 
then it will be as v a : v a :: a b : a b ; 
where, as the ratio of v a to v a is 

known by means of the equidistant ^ b 

rings on the surface, and a b is known, 
a b becomes determined. 



HYDROSTATICS : SPECIFIC GRAVITIES. 301 

Ex. Let d=100 inches, d—5 inches, or the exterior diame- 
ter = 110, and t the thickness = 4 inches. 

Then -0073 t d {v+d) =-0073x4x5 x 105 =-0073 X 2100 
= 15*33 cwt, the required weight of the cylindrical rim. 

Note. — The reader will observe that this process is much 
shorter than those usually employed, even with the aid of ta- 
bles of circles already computed. 

%* See, farther, on kindred subjects, the approximate rules 
in mensuration, pa. 201, &c. 



40 2D 



302 HYDRODYNAMICS. 

CHAPTER XII. 
HYDRODYNAMICS. 

Hydrodynamics is that part of mechanical science which re- 
lates to the motion of non-elastic fluids, and the forces with 
which they act upon bodies. 

This branch of mechanics is the most difficult, and the least 
advanced : whatever we know of it is almost entirely due to 
the researches of the moderns. 

Could we know with certainty the mass, the figure, and the 
number of particles of a fluid in motion, the laws of its motion 
might be determined by the resolution of this problem, viz. to 
find the motion of a proposed system of small free bodies acting 
one upon the other in obedience to some given exterior force. 
We are, however, very far from being in possession of the 
data requisite for the solution of this problem. We shall, there- 
fore, simply present a few of the most usually received theo- 
retical deductions ; and then proceed to state those rules which 
have flown from a judicious application of theory to experi- 
ment. 

Section I. — Motion and Effluence of Liquids. 

1. A jet of water, issuing from an orifice of a proper form, 
and directed upwards, rises, under favourable circumstances, 
nearly to the height of the head of water in the reservoir ; and 
since the particles of such a stream are but little influenced by 
the neighbouring ones, they may be considered as independent 
bodies,inoving initially with the velocity which would be ac- 
quired in falling from the height of the reservoir. And the ve- 
locity of the jet will be the same whatever may be its direction. 

2. Hence, if a jet issue horizontally from any part of the side 
of a vessel standing on a horizontal plane, and a circle be de- 
scribed having the whole weight of the fluid for its diameter, 
the fluid will reach the plane at a distance from the vessel, equal 
to that chord of the circle in which the jet initially moves. 

Thus, if a s be the upper surface of the 
fluid in the vessel, b the place of the ori- 
fice, c f the horizontal plane on which 
the fluid spouts, then c p is equal to e d, 
the horizontal chord of the circle whose 
diameter is a c, passing through b. 



A 


^p 


K. 


A 



HYDRODYNAMICS I EFFLUENCE OF FLUIDS. 



303 



3. When a cylindrical or prismatic vessel empties itself by 
a small orifice, the velocity at the surface is uniformly retarded ; 
and in the time of emptying itself, twice the quantity would be 
discharged if it were kept full by a new supply. 

4. But the quantity discharged is by no means equal to what 
would fill the whole orifice, with this velocity. If the aperture 
is made simply in a thin plate, the lateral motion of the parti- 
cles towards it tends to obstruct the direct motion, and to con- 
tract the stream which has left the orifice, nearly in the ratio 
of two to three. So that in order to find the quantity dis- 
charged, the section of the orifice must be supposed to be di- 
minished from 100 to 62 for a simple aperture, to 82 for a pipe 
of which the length is twice the diameter, and in other ratios 
according to circumstances. 

5. When a syphon, or bent tube, is filled with a fluid, and its 
orifices immersed in the fluids of different vessels, if both sur- 
faces of the fluids are in the same level, the whole remains at 
rest ; but, if otherwise, the longer column of fluid in the sy- 
phon preponderates, and the pressure of the atmosphere forces 
up the fluid from the higher vessel, until the equilibrium is re- 
stored ; and the motion is the more rapid as the difference of 
levels is greater : provided that the greatest height of the tube 
above the upper surface be not more than a counterpoise to the 
pressure of the atmosphere. 

6. If the lower vessel be allowed to empty 
itself, the syphon will continue running as 
long as it is supplied from the upper, and 
the faster, as it descends the further below 
the upper vessel. In the same manner the 
discharge of a pipe, descending from the 
side or bottom of a given vessel, must be 
increased almost without limit by lengthening 
it* 




* An improvement in the construction of the syphon has 
been lately proposed in the Glasgow Mechanics' Magazine, 
and by M. Buntem at Paris. It might be very advantageously 
used if constructed on a large scale, for lowering the water in 
mill dams or canals. The improvement in the present syphon 
is, that the exhausting pipe is enlarged to the same diameter as 
that of the syphon, and its mouth is widened out to something 
of a funnel shape, as in the figure. 

In putting it into action, the short arm is immersed in the 
water as in the usual manner, the bottom of the long arm is 
closed, the exhausting pipe is then filled with water by the fun- 
nel-shaped mouth. On the bottom of the long arm being opened, 
the water flows out, exhausts the air from the syphon, when the 
water, which is wished to be emptied, flows out in a continual 
stream. 



<f? 



(f\ 



S04 hydrodynamics: effluence of fluids. 

7. If a notch or sluice in form of a rectangle be cut in the ver- 
tical side of a vessel full of water, or any other fluid, the quan- 
tity flowing through it will be § of the quantity which would 
flow through an equal orifice placed horizontally at the whole 
depth, in the same time, the vessel being kept constantly full. 

8. If a short pipe elevated in any direction from an aperture 
in a conduit, throw the water in a parabolic curve to the dis- 
tance or range r, on a board, or other horizontal plane passing 
through the orifice, and the greatest height of the spouting fluid 
above that plane, be h, then the height of the head of water 
above that conduit pipe, may be found nearly : viz. by taking 

1st, 2 cot e = ; and 2dly, the altitude of the head a=£ R 

2 H 

X cosec 2 e. 

Example. — Suppose that r=40 feet, and h= 18 feet. Then 
R An 

= — =1-1111111=2 cot 60° 57': and A=§ r xcosec 2 e 

2 h 36 

=20 xcosec 121° 54'=20 X 1'177896 = 23'55792 feet, height 
required. 

Note. — This result of theory will usually be found about | of 
that which is furnished by experiment. 



Section II. — Motion of Water in Conduit Pipes and Open 
Canals, over Weirs, &?c. 

1. When the water from a reservoir is conveyed in long ho- 
rizontal pipes of the same aperture, the discharges made in 
equal times are nearly in the inverse ratio of the square roots 
of the lengths. 

It is supposed that the lengths of the pipes to which this rule 
is applied are not very unequal. It is an approximation not 
deduced from principle, but derived immediately from experi- 
ment. [Bossut, torn. 11, § 647, 648. At § 673, he has given 
a table of the actual discharges of water-pipes, as far as the 
length of 2340 toises, or 14950 feet English.] 

2. Water running in open canals, or in rivers, is accelerated 
in consequence of its depth, and of the declivity on which it 
runs, till the resistance, increasing with the velocity, becomes 
equal to the acceleration, when the motion of the stream be- 
comes uniform. 

It is evident that the amount of the resisting forces can 
hardly be determined by principles already known, and there- 
fore nothing remains but to ascertain, by experiment, the velo- 



HYDRODYNAMICS I PIPES AND CANALS. 305 

city corresponding to different declivities, and different depths 
of water, and to try, by multiplying and extending these expe- 
riments, to find out the law which is common to them all. 

The Chevalier Du Buat has been successful in this re- 
search, and has given a formula for computing the velocity 
of running water, whether in close pipes, open canals, or 
rivers, which, though it may be called empirical, is extremely 
useful in practice. Principes d' Hydraulique. Professor Ro- 
bison has given an abridged account of this book, in his ex- 
cellent article on Rivers and Water-works, in the Encyclopae- 
dia Britannica. 

Let v be the velocity of the stream, measured by the inches it 
moves over in a second ; r a constant quantity, viz. the quotient 
obtained by dividing the area of the transverse section of the 
stream, expressed in square inches, by the boundary or perime- 
ter of that section, minus the superficial breadth of the stream 
expressed in linear inches. 

The mean velocity is that with which, if all the particles were 
to move, the discharge would be the same with the actual dis- 
charge. 

The line r is called by Du Buat the radius, and by Dr. Ro- 
bison the hydraulic mean depth. As its affinity to the radius 
of a circle seems greater than to the depth of a river, we shall 
call it, with the former, the radius of the section. 

Lastly, let s be the denominator of a fraction which expresses 
the slope, the numerator being unity, that is, let it be the quo- 
tient obtained by dividing the length of the stream, supposing 
it extended in a straight line, by the difference of level of its 
two extremities ; or, which is nearly the same, let it be the co- 
tangent of the inclination or slope. 

3. The above denomination being understood, and the section, 
as well as the velocity, being supposed uniform, v in English 
feet, 

_ 307^(R- T y) s >,__ , . . 

s '— ilog(s+H) 

»"=^(r-^ *> 

S *-ilog(s+i=) 
When r and s are very great, 

v = r ^- I _3_ y nearly. 

s 2 — |logs 

The logarithms understood here are the hyperbolic, and are 
found by multiplying the common logarithms by 2*3025851. 

2 d2 



306 HYDRODYNAMICS I PIPES AND CANALS. 



The slope remaining the same, the velocities are as v/r— ^. 

The velocities of two great rivers that have the same decli- 
vity, are as the square roots of the radii of their sections. 

If r is so small, that >/r — T ^ == 0, or r = -Jg-, the velocity 
will be nothing ; which is agreeable to experience ; for in a 
cylindric tube r = h the radius ; the radius, therefore, equals 
two-tenths ; so that the tube is nearly capillary, and the fluid 
will not flow through it. 

The velocity may also become nothing by the declivity be- 
coming so small, that 

™ A-o;but 

s *_aiog( s+ «) 

if — is less than — ■, or than -^th of an inch to an English 

s 500000' 10 & 

mile, the water will have sensible motion. 

4. In a river, the greatest velocity is at the surface, and in 
the middle of the stream, from which it diminishes toward the 
bottom and the sides, where it is least. It has been found by 
experiment, that if from the square root of the velocity in the 
middle of the stream, expressed in inches per second, unity be 
subtracted, the square of the remainder is the velocity at the 
bottom. 

Hence, if the former velocity be = v, the velocity at the bot- 
tom = v — 2 s/v -f- 1. 

5. The mean velocity, or that with which, were the whole 
stream to move, the discharge would be the same with the real 
discharge, is equal to half the sum of the greatest and least ve- 
locities, as computed in the last proposition. 

The mean velocity is, therefore, = v — \/v + h. 
This is also proved by the experiments of Du Buat. 

6. Suppose that a river having a rectangular bed, is increased 
by the junction of another river equal to itself, the declivity 
remaining the same ; required the increase of depth and ve- 
locity. 

Let the breadth of the river == b, the depth before the junc- 
tion d, and after it, x ; and, in like manner, v and v' the mean 

velocities before and after ; then , , is the radius before, 

a h x .u j- c. 3 °7 R 2 • ., 

and i . — the radius after, sov= ; supposing the 

s 

breadth of the river to be such, that we may reject the small 
quantity subtracted from r, in art. 3 ; and, in like manner, 



HYDRODYNAMICS I PIPES AND CANALS. 307 

307 R'^ 

v = 1 • Then, substituting for r and r', we have 

S^ 

307 | b d 

v =— X /—-ii-^, and 

v'^— x J va; . Multiplying these into the 
4 \ v + 2x 



areas 



of the sections b d and h x, we have the discharges, viz. 
307 bd V b d , , , 307 b x s/ b x 

bdv =* — x -j =7 ; and 6 z t* = — x - . 



s 

Now the last of these is double of the former ; therefore, 
b x x/ b x 2 b d s/ b d x 3 4 d 3 

VT+2^ = N/H2rf' 0r H2a;"H2(/' an 

3 d * 4bd * U ♦• t- u i 

# = t— — -—j, a cubic equation which can al- 



b + 2d b + 2 d' 

ways be resolved by Cardan's rule, or by the approximating 
method given at page 92. 

As an example, let b = 10 feet, and d = 1, then x B — f x 
= Y? an d a? = 1*4882, which is the depth of the increased 
river. Hence we have 1*488 x v' = 2 v, and 1*488 : 2 :: v : 
v' } or v : v' :: 37 to 50 nearly. 

When the water in a river receives a permanent increase, the 
depth and the velocity, as in the example above, are the first 
things that are augmented. The increase of the velocity in- 
creases the action on the sides and bottom, in consequence of 
which the width is augmented, and sometimes also, but more 
rarely, the depth. The velocity is thus diminished, till the te- 
nacity of the soil, or the hardness of the rock, affords a sufficient 
resistance to the force of the water. The bed of the river then 
changes only by insensible degrees, and, in the ordinary lan- 
guage of hydraulics, is said to be permanent, though in strict- 
ness this epithet is not applicable to the course of any river. 

7. When the sections of a river vary, the quantity of water 
remaining the same, the mean velocities are inversely as the 
areas of the sections. 

This must happen, in order to preserve the same quantity of 
discharge. (Playfair's Outlines.) 

8. The following table, abridged from Du Buat, serves at 
once to compare the surface, bottom, and mean velocities in 
rivers, according to the principles of art. 4, 5. 



308 



HYDRODYNAMICS I CANALS, RIVERS, &C. 



Velocities of Rivers. 



VELOCITY IN INCHES. 


VELOCITY IN INCHES. 


Surface. 


Bottom. 


Mean. 


Surface. 


Bottom. 


Mean. 


4 


1* 


25 


56 


42-016 


49-008 


8 


3-342 


5-67 


60 


45-509 


52-754 


12 


6-071 


9-036 


64 


49* 


56-5 


16 


9* 


12-5 


68 


52-505 


60-252 


20 


12-055 


16-027 


72 


56-025 


64-012 


24 


15-194 


19-597 


76 


59-568 


67-784 


28 


18-421 


23-210 


80 


63-107 


71-553 


32 


21-678 


26-839 


84 


66*651 


75-325 


36 


25' 


30-5 


88 


70-224 


79-112 


40 


28-345 


34-172 


92 


73-788 


82-894 


44 


31-742 


37-871 


96 


77-370 


86-685 


48 


35-151 


41-570 


100 


81* 


90-5 


52 


38-564 


45'282 









9. The knowledge of the velocity at the bottom is of the 
greatest use for enabling us to judge of the action of the stream 
on its bed. 

Every kind of soil has a certain velocity consistent with the 
stability of the channel. A greater velocity would enable the 
waters to tear it up, and a smaller velocity would permit the 
deposition of more moveable materials from above. It is not 
enough, then, for the stability of a river, that the accelerating 
forces are so adjusted to the size and figure of its channel that 
the current may be in train : it must also be in equilibrio with 
the tenacity of the channel. 

We learn from the observations of Du Buat and others, 
that a velocity of three inches per second at the bottom will 
just begin to work upon the fine clay fit for pottery, and how- 
ever firm and compact it may be, it will tear it up. Yet no 
beds are more stable than clay when the velocities do not 
exceed this : for the water soon takes away the impalpable 
particles of the superficial clay, leaving the particles of sand 
sticking by their lower half in the rest of the clay, which they 
now protect, making a very permanent bottom, if the stream 
does not bring down gravel or coarse sand, which will rub off 
this very thin crust, and allow another layer to be worn off ; a 



HYDRODYNAMICS : CANALS, RIVERS, &C 309 

velocity of six inches will lift fine sand ; eight inches will lift 
sand as coarse as linseed ; twelve inches will sweep along fine 
gravel ; twenty-four inches will roll along rounded pebbles an 
inch in diameter ; and it requires three feet per second at the 
bottom to sweep along shivery angular stones of the size of an 
egg. (Robison on Rivers.) 

10. Mr. Eytelwein, a German mathematician, has devoted 
much time to inquiries in hydrodynamics. In his investiga- 
tions he has paid attention to the mutual cohesion of the liquid 
moleculae, their adherence to the sides of the vessel in which 
the water moves, and to the contrac- 
tion experienced by the liquid vein B r - -r 

when it issues from the vessel under g^ — [j| 
certain circumstances. He obtains for- |^^S 
mulae of the utmost generality, and jglSl L^ 
then applies them to the motion of wa- | T j^^^ 

ter ; 1st, in a cylindric tube ; 2dly, in D *^^ 

an open canal. 

11. Let d be the diameter of the cylindric tube e f, h the 
total height f g of the head of water in the reservoir above the 
orifice f, and I the length e f of the tube, all in inches : then 
the velocity in inches with which the fluid will issue from the 
orifice f will be 



v = 23 i J / + 57 d : ( En S lish measure.] 



this multiplied into the area of the orifice will give the quantity 
per second. 

12. For open canals. Let v be the mean velocity of the cur- 
rent in feet (English), a area of the vertical section of the 
stream, p perimeter of the section, or sum of the bottom and two 
sides, I length of the bed of the canal corresponding to the fall h, 
all in feet : then 



J 958 %-f 



0-109 + 9582 -- + 0-0111 



The experiments of M. Bidone, of Turin, on the motion of 
water in canals, agree within the 80th part of the results of com- 
putations from the preceding formulae. 

1 3. The following table also exhibits Eytelwein 7 s coefficients 
for orifices of different kinds ; their accuracy has, in many cases, 
been amply confirmed. 
41 



310 



HYDRODYNAMICS : EYTELWEIN 7 S RESULTS. 



1 

No. 


Nature of the Orifices employed. 


Ratio between 

the theoretical 

and real 

discharges. 


Coefficients 

for finding 

the velocities 

in Eng. feet 


1 
2 

3 

4 
5 

6 

7 
8 


For the whole velocity due to the 
height 

For wide openings whose bottom 
is on a level with that of the re- 


1 to l'OO 

1 to 0-961 

1 to 0-961 
1 to 0-961 

1 to 0-861 
1 to 0-861 
1 to 0-861 
1 to 0-635 


8-04 

7-7 

7-7 
7-7 

6-9 
6-9 
6-9 
51 


For sluices with walls in a line with 
the orifice 

For bridges with pointed piers . . . 

For narrow openings whose bottom 
is on a level with that of the re- 


For smaller openings in a sluice 


For abrupt projections and square 

piers of bridges 

For openings in sluices without 





14. When a pipe is bent in one or more places, then if the 
squares of the sines of the several changes of direction be added 
into one sum s, the velocity v will, according to Langsdorf, be 

, , , , I 548 d h ,",',"', 

found by the theorem v = |— - — t , ; /, h, d, and v 9 

being all in English inches. 



HYDRODYNAMICS : MOTION IN PIPES, &C. 



311 



8 

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ft 

8 P 



r© 



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Bore 
of the 
Pipes. 


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312 



HYDRODYNAMICS : TABLES, &C. FOR WEIRS. 



Look for the velocity of water in the pipe in the upper row, 
and in the column below it, and opposite to the given diameter 
of the pipe standing in the last column, will be found the per- 
pendicular height of a column or head, in feet, inches, and 
tenths, requisite to overcome the friction of such pipe for 100 
feet in length, and obtain the given velocity. 



Table containing the quantity of water discharged over an 
inch vertical section of a Weir. 



Depth of the 
upper edge 
of the waste- 
board below 
the surface 
in EngUsh 
inches. 


Cubic feet of 
water discharged 
in a minute by 
an inch of the 
wasteboard, ac- 
cording to Du 
Buat's formulae. 


Cubic feet of 
water discharged 
in a minute by an 
inch of the waste- 
board, according 
to experiments 
made in Scotland. 


Gallons x 
of 
282 inches 
correspond- 
ing with 
results in 
col. 3. 


1 


0-403 


0-428 


2-621 


2 


1-140 


1-211 


7*417 


3 


2-095 


2'226 


13-634 


4 


3-225 


3-427 


20-990 


5 


4-507 


4-789 


29-332 


6 


5'925 


6-295 


38-357 


7 


7.466 


7-933 


48-589 


8 


9-122 


9.692 


59*364 


9 


10-884 


11-564 


70-826 


10 


12.748 


13-535 


83-164 


11 


14.707 


15-632 


95-746 


12 


16-758 


17*805 


109-055 


13 


18-895 


20-076 


122-965 


14 


21-117 


22-437 


137-427 


15 


23-419 


24-883 


152-408 


16 


25-800 


27*413 


167-905 


17 


28-258 


30-024 


183-897 


18 


30-786 


32-710 1 


200-350 1 



To the above table, originally due to Du Buat, is added a 
third column, containing the quantities of water discharged, as 
inferred from experiments made in Scotland, and examined 
by Dr. Robison, who found that they in general gave a dis- 
charge -jJg- greater than that which is deduced from Du Buat's 
formulae. We would recommend it therefore to the engineer 
to employ the third column in his practice, or the fourth if he 
wish for the result in gallons. , 



HYDRODYNAMICS : DISCHARGES OVER WEIRS. 313 

If they be odd quarters of an inch, look in the table for as 
many inches as the depth contains quarters, and take the eighth 
part of the answer. Thus, for 3§ inches, take the eighth part 
of 24-883, which corresponds to 15 inches. This is 3*110. 

15. The quantity discharged increases more rapidly than the 
width : to obtain a correct measure of it, if n be the width or 
length of the wasteboard in inches, take (n + ^\ n) times the 
quantity for one inch of wasteboard of the given depth, from the 
preceding table. 

In the preceding table it is supposed that the water from 
which the discharge is made is perfectly stagnant; but if it 
should happen to reach the opening with any velocity, we have 
only to multiply the area of the section by the velocity of the 
stream. 

16. When the quantity of water q discharged over a weir is 
known, the depth of the edge of the wasteboard, or h, may be 
approximated from the following formulae, / length of waste- 
board 

2 



H = Gts™)* = (hit) 7 near1 ^ 



Reciprocally, q = 11$ / . h ¥ nearly : or, more accurately by 
adding the correction in article 15. 

17. The quantities discharged for any given width are as the 

I power of the depth, or as h 2 . 

Hence, to extend the use of the table to greater depths, we 
have only for 

Twice any depth, take q x 2*828 

3 times q x 5*196 

4 times q x 8*000 

5 times q x 11*180 

6 times q x 14*697 

7 times q x 18*520 

8 times q X 22-627 

9 times q X 27*000 

10 times q X 31.623 

and the results will be nearly true. 

To make them still more correct, where great accuracy is 
required, add to them their thousandth part. 

2E 



314 HYDRODYNAMICS : DISCHARGES OP WATER. 



Examples of the use of the Tables and Rules. 

Ex. 1. Let the depth be 10 inches below the upper surface of 
the water, and the width 8 inches. How many cubic feet of 
water will be discharged in a minute ? 

cub. feet. 

By table q to depth 10, width 1 = 13-535 
Multiply this by n = 8 

106-280 
Add gV °f this product 5*314 

Discharge in one minute = 111*594 

Ex. 2. Let the depth be 9 feet, and the width 1 foot. Re- 
quired the cubic feet discharged in a minute. 

By table q to depth 12 inch, width 1 inch = 17*805 
Factor for 9 times depth is 27 = 3 x 9 3 

53*415 
9 

Quantity for 1 inch width 480*735 

Multiply by n = 12 

5768*802 
Add ^ of the product 286*441 

Total quantity in cubic feet = 6055-261 

Ex. 3. Let a square orifice of 6 inches each side be placed in 
a sluice-gate with its top 4 feet below the upper surface of the 
water : how much will it discharge in a minute ? 

Here the quantity discharged by a slit in depth 48 inches, 
must be taken from one in depth 54 inches. 

cub. feet. 
3 

For 54, multiply q at 6 by 3 2 or 27 169*965 

For 48, q at 12 by 42 or 8 142*440 

Difference 21-525 

21-525 X (6 + /J = 173*4 cubic feet, quantity discharged. 



HYDRODYNAMICS I DISCHARGES THROUGH SLUICES, &C. 315 

Note. — In an example like this, it is a good approximation, 
to multiply continually together the area of the orifice, the 
number 336,* and the square root of the depth in feet of the 
middle of the orifice. 

Thus, in the preceding example, it will be \ X h X 336 
X s/4-25=iX 336x2-062 = 173*2 cubic feet. 

The less the height of the orifice compared with its depth un- 
der the water, the nearer will the result thus obtained approach 
to the truth. 

If the height of the orifice be such as to require considera- 
tion, the principle of Art. 7 of the preceding section may be 
blended with this rule. 

Thus, applying this rule to Ex. 2, we shall have area x 
s/depthx 336x| = 9x3x224=6048, for the cubic feet dis- 
charged. This is less than the former result by about its 900th 
part. It is, therefore, a good approximation, considering its 
simplicity : it may in many cases supersede the necessity of re- 
currence to tables. 

18. Prob. Given the vertical section of a river or other 
stream, to determine the swell occasioned by the piers of a 
bridge, or the sides of a cleaning sluice which contract the 
passage by a given quantity, for a short length only of the 
channel ; the velocity of the stream being also known. Let 
v be the velocity of the stream, independently of the effect 
of the bridge, r the section of the river, and a the amount 
of the sections between the piers ; let 2 g, instead of being 
taken at 64§, be reckoned 58*6, to accord with the effect of ex- 
perimental contractions through arches of bridges, &c, and let 
s be the slope of the bed of the river, or the sine of its an- 
gle with the horizon ; then Du Buat (torn. i. p. 225) gives for 
the swell or rise of the stream in feet, which will be occasioned 

by the obstruction f h s \ ( f — "1 — 1 ) : 

a theorem, by means of which we may approximate to the said 
swell in any proposed case. 

The value of s will, of course, be different in different cases ; 
but if we assume ^V? or '05, as a mean value, it will enable us 
to compute and tabulate results, which, though they cannot be 
presented as perfectly correct, may be regarded as exhibiting a 
medium between those that will usually occur ; and will serve 
to anticipate the consequences of floods of certain velocities, 
when constrained to pass through bridges which more or less 
contract the stream. 

* 336 = 5-6x60. 



316 hydrodynamics: swells occasioned by bridges. 



Table of the Rise of Water occasioned by Piers of Bridges 
and other Contractions.* 



p o 

§£ 

o u 

. <v 

m 

> 


Amount of obstructions, compared with the vertical section of the River. 




1-lOth. 


2-10ths. 


3-10ths. 


4-10ths. 


5-10ths. 


6-10ths 


7-10ths. 


8-10ths. 


9-10ths. 


Proportional Rise of Water, in feet and decimals. 




feet 


feet 


feet 


feet 


feet 


feet 


feet 


feet 


feet 


1 


00157 


00377 


0-0698 


0-1192 


0-2012 


03521 


0-6780 


1-6094 


6-6389 




2 


0277 


0665 


01231 


0-2102 


0-3548 


6208 


1-1995 


2-8378 


117058 


r Ordinary 


3 


00477 


01144 


0-2118 


0-3618 


6107 


1-0687 


20580 


4-8850 


201504 


4 


0760 


0-1822 


0-3372 


0-5759 


9719 


1-7008 


3-2755 


7-7750 


320720 


C floods. 


5 


01165 


0-2793 


0-5168 


0-8782 


1-4895 


26066 


5-0202 


11-9160 


49-1535 


f Violent 


6 


0-1558 


0-3736 


6912 


1-1807 


1-9925 


3-4S68 


6-7154 


15-9398 


65-7518 


7 


0-2078 


0-4983 


0-9221 


1-5750 


2-6578 


46511 


8-9578 


21-2626 


87-7080 


( floods. 


8 0-2678 


0-6423 


1-1884 


20299 


3-4255 


5-9947 


11-5154 


27-4042 


113-0422 


; Unusually 


9 0-3359 


0-80.54 


1-4903 


2-5566 


4-2956 


7-5172 


14-4777 


34-3646 


141-7541 


> violent 


10 1 0-4119 


0-9877 


1-8276 


3-1218 


5-2680 


9-2190 


17-7551 


42-1440 


173-8440 


) floods. 



19. It will be evident, from an inspection of this table, that 
even in the case of ordinary floods, old bridges with piers and 
starlings, occupying 6 or 7 tenths of the section of the river, 
will produce a swell of 2, 3, or more feet, often overflowing 
the river's banks and occasioning much mischief. Also, that 
in violent floods, an obstruction amounting to 7 tenths of the 
channel will cause a rise of 7 or 8 feet, probably choking 
up the arches, and occasioning the destruction of the bridge. 
Greater velocities and greater contractions produce a rapid 
augmentation of danger and mischief ; as the table obviously 
shows. 

20. The same principles and tabulated results serve to esti- 
mate the fall from the higher to the lower side of a bridge, on 
account of an ebbing tide, &c. Thus, for old London Bridge, 
where the breadth of the Thames is 926 feet, and the sum of 
the water ways at low water only 236 feet ; the amount of ob- 
structions was 690 feet, about 7§ tenths of the entire section ; 
so that a velocity of 3i feet per second would give a fall of 
nearly 4| feet, agreeing with the actual result. 

At Rochester Bridge, before the opening of the middle 
arches, the piers and starlings presented an obstruction of 
seven-tenths, and at the time of greatest fall, the velocity 

* A similar table was computed by Mr. Wright, of Durham, more than 50 years 
ago, and inserted in the first edition of Dr. Hutton's treatise on Bridges ; but it is 
not constructed upon a correct theory. 



HYDRODYNAMICS : FALL UNDER ARCHES. 317 

100 yards above bridge exceeded 6 feet per second. This, 
from the table, would occasion a fall of more than 6-7 feet ; and 
the recorded results vary from 6i to 7 feet. 

At Westminster Bridge, where the obstructions are about 
one-sixth of the whole channel, when the velocity is 2§ feet, 
the fall but little exceeds half an inch : a result which the 
table would lead us to expect. 



Section III. — Contrivances to measure the velocity of 
running waters. 

1. For these purposes, various contrivances have been pro- 
posed, of which two or three may be here described. 

Suppose it be the velocity of the water of a river that is 
required ; or, indeed, both the velocity and the quantity 
which flows down it in a. given time. Observe a place where 
the banks of the river are steep and nearly parallel, so as to 
make a kind of trough for the water to run through, and by 
taking the depth at various places in crossing make a true 
section of the river. Stretch a string at right angles over it, 
and at a small distance another parallel to the first. Then 
take an apple, an orange, or other small ball, just so much 
lighter than water as to swim in it, or a pint or quart bottle 
partly filled with water, and throw it into the water above the 
strings. Observe when it comes under the first string, by 
means of a quarter second pendulum, a stop watch, or any 
other proper instrument ; and observe likewise when it arrives 
at the second string. By this means the velocity of the upper 
surface, which in practice may frequently be taken for that of 
the whole, will be obtained. And the section of the river at 
the second string must be ascertained by taking various depths, 
as before. If this section be the same as the former, it may be 
taken for the mean section : if not, add both together, and 
take half the sum for the mean section. Then the area of the 
mean section in square feet being multiplied by the distance 
between the strings in feet, will give the contents of the water 
in solid feet, which passed from one string to the other during 
the time of observation : and this by the rule of three may be 
adapted to any other portion of time. The operation may 
often be greatly abridged by taking notice of the arrival of the 
floating body opposite two stations on the shore, especially 
when it is not convenient to stretch a string across. An arch 
42 2 E 2 



318 HYDRODYNAMICS : STREAM MEASURERS. 

of a bridge is a good station for an experiment of this kind, be- 
cause it affords a very regular section and two fixed points of 
observation ; and in some instances the sea practice of heaving 
the log may be advantageous. Where a time-piece is not at 
hand, the observer may easily construct a quarter second or 
other pendulum, by means of the rules and table relating to 
pendulums, in the Dynamics. 

2. M. Pitot invented a stream measurer of a simple con- 
struction, by means of which the velocity of any part of a 
stream may readily be found. This instrument is composed 
of two long tubes of glass open at both ends ; one of these tubes 
is cylindrical throughout ; the other has one of its extremities 
bent into nearly a right angle, and gradually enlarges like a 
funnel, or the mouth of a trumpet : these tubes are both fixed 
in grooves in a triangular prism of wood ; so that their lower 
extremities are both on the same level, standing thus one by 
the side of the other, and tolerably well preserved from ac- 
cidents. The frame in which these tubes stand is graduated, 
close by the side of them, into divisions of inches and lines. 

To use this instrument, plunge it perpendicularly into the 
water, in such manner that the opening of the funnel at the 
bottom of one of the tubes shall be completely opposed to the 
direction of the current, and the water pass freely through the 
funnel up into the tube. Then observe to what height the 
water rises in each tube, and note the difference of the sides ; 
for this difference will be the height due to the velocity of the 
stream. It is manifest, that the water in the cylindrical tube 
will be raised to the same height as the surface of the stream, 
by the hydrostatic pressure : while the water entering from the 
current by the funnel into the other tube, will be compelled to 
rise above that surface by a space at which it will be sustained 
by the impulse of the moving fluid : that is, the momentum of 
the stream will be in equilibrio with the column of water sus- 
tained in one tube above the surface of that in the other. In 
estimating the velocity by means of this instrument, we must 
have recourse to theory as corrected by experiments. Thus, if 
h, the height of the column sustained by the stream, or the 
difference of heights in the two tubes, be in feet, we shall have 
v = 6'5 y/ h, nearly, the velocity, per second, of the stream ; 
if h be in inches, then v — 22*47 >/ h, nearly : or farther ex- 
periments made with the same instrument may a little modify 
these co-efficients. 

It will be easy to put the funnel into the most rapid part of 
the stream, if it be moved about to different places until the dif- 
ference of altitude in the two tubes becomes the greatest. In 
some cases it will happen, that the immersion of the instrument 



HYDRODYNAMICS I STREAM MEASURERS. 319 

will produce a little eddy in the water, and thus disturb the 
accuracy of the observation : but keeping the instrument im- 
mersed only a few seconds will correct this. The wind would 
also affect the accuracy of the experiments ; it is, therefore, 
advisable to make them where there is little or no wind. By 
means of this instrument a great number of curious and useful 
observations may easily be made : the velocity of water at 
various depths in a canal or river may be found with tolerable 
accuracy, and a mean of the whole drawn, or they may be 
applied to the correcting of the theory of waters running down 
gentle slopes. The observations may likewise be applied to 
ascertain whether the augmentations of the velocities are in 
proportion to the increase of water passing along the same 
canal, or what other relations subsist between them, &c. 

Where great accuracy is not required, the tube, with the 
funnel at bottom, will alone be sufficient ; as the surface of the 
water will be indicated, with tolerable precision, by that part 
of the prismatic frame for the tube which has been moistened 
by the immersion : and the velocities may be marked against 
those altitudes in the tube which indicate them. 

3. Another good and simple method of measuring the ve- 
locity of water in a canal, river, &c. is that described by the 
Abbe Mann, in his treatise on rivers ; it is this : — Take a cy- 
lindrical piece of dry light wood, and of a length something 
less than the depth of the water in the river ; about one end 
of it let there be suspended as many small weights as may 
keep the cylinder in a vertical or upright position, with its 
head just above water. To the centre of this end fix a 
small straight rod, precisely in the direction of the cylinder's 
axis : in order that, when the instrument is suspended in the 
water, the deviations of the rod from a perpendicularity to the 
surface of it, may indicate which end of the cylinder goes 
foremost, by which may be discovered the different velocities 
of the water at different depths ; for when the rod inclines 
forward, according to the direction of the current, it is a 
proof that the surface of the water has the greatest velocity ; 
but when it reclines backward, it shows that the swiftest 
current is at the bottom ; and when it remains perpendicular, 
it is a sign that the velocities at the top and bottom are 
equal. 

This instrument, being placed in the current of a river or 
canal, receives all the percussions of the water throughout the 
whole depth, and will have an equal velocity with that of the 
whole current from the surface to the bottom at the place 
where it is put in, and by that means may be found, both 



320 



HYDRODYNAMICS : STREAM MEASURERS. 



with exactness and ease, the mean velocity of that part of the 
river for any determinate distance and time. 

But to obtain the mean velocity of the whole section of the 
river, the instrument must be put successively both in the mid- 
dle and towards the sides, because the velocities at those places 
are often very different from each other. Having by these 
means found the several velocities, from the spaces run over in 
certain times, the arithmetical mean proportional of all these 
trials, which is found by dividing the common sum of them 
all by the number of the trials, will be the mean velocity of 
the river or canal. And if this medium velocity be multiplied 
by the area of the transverse section of the waters of any place, 
the product will be the quantity running through that place in 
a second of time. 

The cylinder may be easily guided into that part which we 
want to measure, by means of two threads or small cords, 
which two persons, one on each side of the canal or river, must 
hold and direct ; taking care at the same time neither to retard 
nor accelerate the motion of the instrument. 

4. Let a a' b b' be a hollow cylinder, open at both ends, and 
let it be capable of being fixed by the side of a platform or of 
a boat, so that its lower extremity b b' may be placed at any 
proposed depth below h r, the upper sur- 
face of the stream. Let p p' be pulleys, 
fixed at opposite sides of the top and bot- 
tom of the tube. To g, a globe of specific 
gravity nearly the same as that of water, 
let a cord g p' p s be attached, passing 
freely over the pulleys p', p, and having 
sufficient length towards s to allow of its 
running off to any convenient distance. 
Then, the bottom of the tube being immersed to any proposed 
depth, let the globe g be exposed to the free operation of 
the stream ; and as it is carried along with it, it will in 1,2, 5, 
or 10 seconds, or any other interval of time, draw off from a 
fixed point, as s, a portion of cord ; from which and the 
time elapsed, the velocity at the assigned depth will become 
known. 

Mr. Saumarez invented, in 1720, an instrument called the 
Marine Surveyor, for the double purpose of measuring a 
ship's way, and ascertaining the velocities of streams. It is 
described in the Phil. Trans, vol. 33 ; and in the succeeding 
volume a curious example of its use is given in " tables 
showing the strength and gradual increase and decrease of 
the tides of flood and ebb in the river Thames, as observed 



S p 

A r 

H 


A' 

R 


mm 


mi 


?5=S7^Hh: 


— ~ *— ^ 






nsrtii 


i^BUli 






s=H 


I^MH 



HYDRODYNAMICS ! OLD LONDON BRIDGE. 321 

in Lambeth Reach." They are too extensive to be inserted 
here ; but are truly interesting, and may be seen in Phil. 
Trans. Abridged, vol. vii. p. 133. 



Section IV. — Effects of the Old London Bridge on the 
Tides, fyc. 

In the first volume of Dr. Hutton's Tracts, 8vo. there are 
inserted some curious papers drawn up by Mr. Robertson, of 
Christ's Hospital, and others, on London Bridge, and on the 
probable consequences, in reference to the tides, of erecting 
a new bridge across the Thames, viz. Blackfriars. Such docu- 
ments are not only interesting as matters of scientific history, 
but become valuable in process of time ; as the comparison of 
facts with theoretic predictions is subservient to the correction 
of the theory itself. With a similar object in view, I here 
introduce abridged accounts of some valuable facts with re- 
gard to the motion and level of the tides in the Thames at 
London, collected in 1820 and 1821, when the project of a new 
bridge of five arches, instead of the old bridge, originally of 20 
arches, and a very contracted water-way, came first under con- 
sideration. 

LONDON BRIDGE. 

Result of the levels of tides observed from 23d September 
to 25th October, 1820, between the entrance of the London 
Docks and Westminster Bridge. Also the transverse sections 
of the river Thames at London Bridge, Southwark Bridge, 
Blackfriars Bridge, Waterloo Bridge, and Westminster Bridge, 
collected from four drawings of the above surveys, made by 
Mr. Francis Giles, under the direction of Mr. James Mon- 
tague, pursuant to an order of the Select Committee of Bridge 
House Lands, of the 8th September, 1820. 

London Docks to London Bridge. 

The high water of spring tides at the entrance of the London 
Docks, averaged a level of 1.5 inch higher and 10 minutes 
earlier time, than at the lower side of London Bridge. 

The low water of ditto at ditto averaged a level of 3 inches 
lower, and 9 minutes earlier time, than at ditto. 

The high water of neap tides at ditto averaged a level of one 
inch, and 8 minutes earlier time, than at ditto. 

The low water of ditto at ditto averaged a level of 2 inches 
lower, and 14 minutes earlier time, than at ditto. 



322 HYDRODYNAMICS : EFFECTS OF LONDON BRIDGE. 



London Bridge, 

High water of the highest spring tides occurs at three or four 
o'clock. — High water of the lowest neap tides occurs at eight 
or nine o'clock. 

Spring tides flow four or five hours, and ebb seven to eight 
and a half hours. — Neap tides flow five to five and a half hours, 
and ebb six and a half to eight hours. 
The high water of spring tides produced an average fall through 

London Bridge of 8 inches, but the greatest fall at high water 

was 1 foot 1 inch. — October 24th. 
The low water of ditto, through ditto, of 4 feet 4 inches, but 

the greatest fall at low water was 5 feet 7 inches. — Septem- 
ber 27th. 
The high water of neap tides through ditto of 5 inches. 
The low water of ditto, through ditto, 2 feet 1 inch, but the 

least fall at low water was 1 foot 1 inch. — October 16th. 

The flood of spring tides of October 21st and 23d, produced 
slack water through the bridge in about 40 minutes after low 
water below bridge, from which time a head gradually in- 
creased (below bridge) to 1 foot 10 inches at half flood, and 
then regularly increased to about 8 inches at high water. — The 
first flow of these tides, nevertheless, began above bridge about 
20 minutes after low water time below bridge, although the 
water was then about 2 feet 6 inches higher above than below 
bridge ; the time of low water below bridge averaged 10 
minutes earlier than above bridge. 

The ebb of these tides produced slack water at the bridge 
about 30 minutes after high water, and then gradually sunk to 
their greatest fall at low water. — The time of high water of Oc- 
tober 21st and 23d, was the same below as above bridge ; but 
the average time of high water spring tides is 9 minutes earlier 
below than above bridge. 

The flood of neap tide, October 30th, produced slack water 
through the bridge, in about two hours after low water time be- 
low bridge (when there was some land flood in the river), from 
which time a head gradually increased (below bridge) to 1 foot 
3 inches at two-thirds flood, and then regularly decreased to 4 
inches at high water. — The first flow of this tide, nevertheless, 
began above bridge, about 1 hour after low water time below 
bridge, although the water was then 1 foot higher above than 
below bridge ; but the average time of low water below bridge 
is 32 minutes earlier than above bridge. 

The ebb of this tide produced slack water at the bridge 



HYDRODYNAMICS : EFFECTS OF LONDON BRIDGE. 



323 



about 15 minutes after high water above bridge, and then gra- 
dually sunk to its greatest fall at low water. — The time of high 
water of October 30th, was 15 minutes earlier below than above 
bridge, and the average time of high water neap tides is 15 
minutes earlier below than above bridge. 



London Bridge to Westminster Bridge. 

The high water line from the upper side of London Bridge 
to Westminster Bridge is generally level, unless influenced by 
winds and land floods. 

The time of high water is about 10 minutes earlier at Lon- 
don than Westminster Bridge. 



The mean low 
water line has 

a fall of 
Ditto 
Ditto 
Ditto 



in. pts. min. 

4 from Westminster to Waterloo Bridge, time 7 



4 3 from Waterloo to Blackfriars Bridge 
3 2 from Blackfriars to Southwark Bridge 
5 from Southwark to London Bridge 



Total foot 10 



- 5 

. 4 



22 



later at Westmin- 
ster than at Wa- 
terloo Bridge. 

Ditto. 

Ditto. 

Ditto. 



Areas of the Transverse Sections in the River Thames at 
London Bridge. 



At an extraordinary high water or level of 2 feet above"^ 
the average spring tide high water mark, at the I 
Hermitage entrance to the London Docks, as set- > 
tied by the Corporation of the Trinity House, Au- 
gust, 1800. J 
At the Trinity High Water mark, or Datum 

Under the Datum 
At an average"} ft. in. 

Spring Tide C below Bridge, or Level of 6*5 
High Water. 3 
At Ditto Spring") 

Tide High C above Bridge, or Level of 1 2*0 
Water. 3 
At Ditto Neap"} 
Tide High C above Bridge, or Level of 4 3-0 
Water. 3 
At Ditto Spring "} 
and Neap Tide C above Bridge, or Level of 14 5-0 

Low Water, j 

At Ditto Neap"} 

Tide Low C below Bridge, or Level of 16 5*0 

Water. 3 

At Ditto Spring"} 

Tide Low C below Bridge, or Level of 18 9-0 
Water. 3 ' 



Feet. 
8,130 


Abate for 
Water- 
works. 
Feet. 
1,000 


Feet. 
7,130 


7,360 






7,122 


850 


6,272 


6,837 


810 


6,027 


5,293 


600 


4,693 


1,488 




1,488 


1,030 




1,030 


540 




540 



324 



HYDRODYNAMICS I EFFECTS OF LONDON BRIDGE. 



At Southwark, Blackfriar's, Waterloo, and Westminster 

Bridges. 

At the above level of extraordinary 

high water ..... 
At the Trinity high water, mark, or 

datum . . . . 

At an average Spring tide high water, 

or level of 1 foot 2 inches under 

the ditto ... 
At ditto Neap tide high water, or level 

of 4 feet 3 inches under the ditto . 
At ditto Spring and Neap tide low 

water 

London, 12th March, 1821. — (Published in a letter addressed to G. H. Sumner, 
Esq. M. P. by a scientific architect.) 

Gradation of the Ebbing and Flowing of the tide at Lon- 
don Bridge, taken above and below, on the 29th of July, 
1821 ; being the day of the new moon. 
ABOVE BRIDGE. 



Southwark 
Bridge. 


Blackfriars 
Bridge. 


Waterloo 
Bridge. 


Westminster 
Bridge. 


15,260 


15,460 


19,822 


16,750 


13,940 


14,117 


17,707 


15,198 


13,170 


12,975 


16,447 


14,015 


11,135 


10,590 


13,116 


11,380 


5,012 


3,724 


3,382 


3,720 



Low Water at Pepper Alley, 50 
minutes past nine o'clock in the 
morning. 



FLOOD TIDE. 

Depth of water when flood com- 
menced 
1st hour rise 
2d hour 
3d hour 
4th hour 
45 minutes 



Ft. In. 



4 hours and 45 minutes'*' : 18 5 



BELOW 

Low Water at Coxes' Quay, 30 mi- 
nutes past nine o'clock in the morn- 
ing. 

FLOOD TIDE. 



Sigh Water at Pepper Alley, 35 mi- 
nutes past two o'clock in the after- 
noon. 



EBB TIDE. 

1st hour fall 
2d hour 
3d hour 
4th hour 
5th hour 
6th hour 
7th hour 
55 minutes 

Depth at low water 



Ft. In. 
2 1 



11 
5 6 



Depth of water when flood com- 
menced 
1st hour rise 
2d hour 
3d hour 
4th hour 
48 minutes 



Ft. In. 



4 hours and 48 minutes . 18 10 



7 hours and 55 minutes . 18 5 

BRIDGE. 

High Water at Coxes' Quay, 18 mi- 
nutes past two o'clock in the after- 
noon. 



1st hour fall . 
2d hour 
3d hour 
4th hour 
5th hour 
6th hour 
7th hour 
59 minutes . 
Depth left 



Ft. In. 
2 1 
4 



4 
3 1 

2 7 
2 3 
1 9 
1 6 
11 
4 



7 hours and 59 minutes . 18 10 

* # * The object of this statement was to show, that the old bridge tended to 
retain the water above bridge and assist the navigation up the river. 



HYDRODYNAMICS : WATERMILLS. 325 



Difference between the Levels of High and Low Water Spring 
Tides, between Rotherhithe and Batter sea in the year 1820. 

/• i. 

Rotherhithe, Old Horse Ferry - - - 21-10 

London Old Bridge 18-2 

Blackfriars - 14-9 

Westminster - - - - - - -12-6 

Vauxhall 12-2 

Battersea - - - - - - -11-6 

From Battersea Bridge to London Bridge, 5 miles ; from London Bridge to Old 
Horse Ferry, 1£ miles. From London Bridge to the Nore, 44 miles.* 



Section V. — Watermills. 



1. The impulse of a current of water, and sometimes its 
weight and impulse jointly, are applied to give motion to mills 
for grinding corn and for various other purposes. Sometimes 
the impulse is applied obliquely to floatboards in a manner 
which may be comprehended at once by reference to a smoke- 
jack. In that, the smoke ascends, strikes the vanes obliquely, 
and communicates a rotatory motion. Imagine the whole 
mechanism to be inverted, and water to fall upon the vanes, 
rotation would evidently be produced ; and that with greater 
or less energy in proportion to the quantity of water and the 
height from which it falls. 

Water-wheels of this kind give motion to mills in Ger- 
many, and some other parts of the continent of Europe. I 
have, also, seen mills of the same construction in Balta, the 
northernmost Shetland Isle. But wherever they are to be 
found, they indicate a very imperfect acquaintance with 
practical mechanics ; as they occasion a considerable loss of 
power. 

2. Water frequently gives motion to mills, by means of 
what is technically denominated an undershot wheel. This 
has a number of planes disposed round its circumference, 
nearly in the direction of its radii, these floatboards (as they 
are called) dipping into the stream, are carried round by it ; as 
shown in the accompanying diagram. The axle of the wheel, of 

* The preceding results will always be valuable, as they supply striking evidence 
of the effect of a broken dam, such as many of our old bridges present. I regret 
that the contrast occasioned by the large arches of the new bridge cannot yet be 
presented : for though the old bridge is removed, the entire obstructions occasioned 
by the starlings were not taken away when this volume went to the press. 
43 2 F 




326 HYDRODYNAMICS I WATERMILLS. 

course, by the intervention of pro- 
per wheels and pinions, turns the 
machinery intended to be moved. 
Where the stream is large and un- 
confined, the pressure on each float- 
board is that corresponding to the 
head due to the relative velocities 
(or difference between the veloci- 
ties of stream and floatboard) : this 
pressure is, therefore, a maximum when the wheel is at rest ; 
but the work performed is then nothing. On the other hand, 
the pressure is nothing when the velocity of the wheels equals 
that of the stream. Consequently, there is a certain interme- 
diate velocity, which causes the work performed to be a maxi- 
mum. 

The weight equal to the pressure is q (<S h — >/ h) 2 9 q 
being the quantity of water passing in a second, h the height 
due to v the velocity of the water, and h that due to u the 
velocity of the floatboard. Considering this as a mass attached 
to the wheel, its moving force is obtained by multiplying 
it into u : and as x/ h — >/ h varies as v — u, this mov- 
ing force varies as (v — u) 2 u which is a max. when u = |v. 
In this case, then, the rim of the wheel moves with | of the 
velocity of the stream ; and the effect which it produces is 

Q X (| v) 2 X I V = 2T Q v 3 : 
so that the undershot wheel, according to the usual theory, 
performs work = -^ Y of the moving force. 

Friction, and the resistance of fluids, modify these results ; 
but Smeaton and others have found that the maximum work 
is always obtained when u is between h v and \ v. 

3. Where the floats are not totally immersed, the water is 
heaped upon them ; and in this case the pressure is that due to 

2 H. 

4. When the floatboards move in a circular sweep close 
fitted to them, or, in general, when the stream cannot escape 
without acquiring the same velocity as the wheel, the cir- 
cumstances on which the investigation turns become analo- 
gous to what happens in the collision of non-elastic bodies. 
The stream has the velocity v before the stroke which is re- 
duced to u, and the quantity of motion corresponding to the 
difference, or to v — u, is transferred to the wheel ; this turns 
with the velocity u ; and therefore the effect of the wheel is as 

\ u, or, — — ; which is a maximum whenv = 2 u ; 

v / v 

being then i of the moving power. 



HYDRODYNAMICS I WATERMILLS. 



32' 




Hence appears the utility of constraining the water to 
move in a narrow channel. 

5. The undershot wheel is used where a large quantity of 
water can be obtained with a moderate fall. But where the 
fall is considerable the overshot is almost always employed. 
Its circumference is formed into 
angular buckets, into which the 
water is delivered either at the 
top or within 60° of it : 52° § is 
the most advantageous distance. 
In that case, if r=the full radius 
of the wheel, h the whole, and h 
the effective height of the fall, 
h = r (1 + sin 37° i) = 1*605 r, 
and r = '623 h. If the friction 
be about | of the moving power, 

v, the velocity of the circumf. of the wheel to produce a maxi- 
mum effect, will = 2*67 \/ h. Here, too, a fall of } h will 
give the water its due velocity of impact upon the wheel : and 

3 

122*176 s h 2 lbs = the mechanical effect, s being the section, in 
feet, of the stream that supplies the buckets. 

Mr. Smeaton's experiments led him to conclude that over- 
shot wheels do most work when their circumferences move at 
the rate of 3 feet in a second, and that when they move con- 
siderably slower than this, they become unsteady and irregular 
in their motion. This determination is, however, to be un- 
derstood with some latitude. He mentions a wheel 24 feet 
in diameter, that seemed to produce nearly its full effect, 
though the circumference moved at the rate of 6 feet in a 
second ; and another of the diameter of 33 feet, of which the 
circumference had only a velocity of 2 feet in a second, with- 
out any considerable loss of power. The first wheel turned 

round in 12 .6, the latter in 51 .9. 

6. Where the fall is too small for an overshot wheel, it is 
most advisable to employ a breast-wheel (such as exhibited in 
the margin), which partakes of its properties ; its floatboards 
meeting at an angle, so as to be assimilated to buckets, and 
the water being considerably confined 
within them by means of an arched 
channel fitting moderately close, but 
not so as to produce unnecessary fric- 
tion. But when the circumstances 
do not admit of a breast-wheel, then 
recourse must be had to the under- 
shot For such a wheel it is best, 




328 



HYDRODYNAMICS : WATERMILLS. 




that the floatboards be so placed as to be perpendicular to the 
surface of the water at the time they rise out of it ; that only 
one half of each should ever be below the surface, and that 
from 3 to 5 should be immersed at once. The Abbe Mann pro- 
posed that there should not be more than six or eight float- 
boards on the whole circumference. 

7. Mills moved by the reaction of water are usually deno- 
minated Barker's Mills; sometimes, however, Parent's ; at 
others, Segner's. But the invention is doubtless Dr. Barker's. 

In the marginal diagram, where c d is a 
vertical axis, moving on a pivot at d, and 
carrying the upper millstone m, after pass- 
ing through an opening in the fixed mill- 
stone c. Upon this axis is fixed a vertical 
tube t t communicating with a horizontal 
tube a b, at the extremities of which A,B,are 
two apertures in opposite directions. When 
water from the mill-course m n is intro- 
duced into the tube t t, it flows out of the apertures, A, b, and 
by the reaction or counter-pressure of the issuing water the 
arm a b, and consequently the whole machine, is put in motion. 
The bridge-tree a b is elevated or depressed by turning the nut 
c at the end of the lever c b. 

In order to understand how this motion is produced, let us 
suppose both the apertures shut, and the tube t t filled with 
water up to t. The apertures a b, which are shut up, will be 
pressed outwards by a force equal to the weight of a column of 
water whose height is t t, and whose area is the area of the 
apertures. Every part of the tube a b sustains a similar pres- 
sure ; but as these pressures are balanced by equal and opposite 
pressures, the arm a b is at rest. By opening the aperture at 
a, however, the pressure at that place is removed, and conse- 
quently the arm is carried round by a pressure equal to that of 
a column t t, acting upon an area equal to that of the aperture a. 
The same thing happens on the arm t b ; and these two pres- 
sures drive the arm a b round in the same direction. This ma- 
chine may evidently be applied to drive any kind of machinery, 
by fixing a wheel upon the vertical axis c d. 

8. Mr. Rumsey, an American, and Mr. Segner, improved 
this machine, by conveying the water from the reservoir, not 
by a pipe, in greater part of which the spindle turns, but by a 
pipe which descends from a reservoir, as p, until it reaches 
lower than the arms a b, and then turns up by a curvilinear 
neck and collar, entering between the arms at the lower part, as 
shown in the figure. This greatly diminishes the friction. 

9. Professor Playfair has correctly remarked that the 



HYDRODYNAMICS : WATERMILLS. 329 

moving force becomes greater, after the machine has begun to 
move ; for the water in the horizontal arms acquires a centri- 
fugal force, by which its pressure against the sides is increased. 
When the machine works to the greatest advantage, the 
centre of the perforations should move with the velocity 

—\^hg" where r is the radius of the horizontal arm, measured 

from the axis of motion to the centre of the perforation, and r' 
the radius of the perpendicular tube, g being put for the force 
of gravity, or 32} feet. 

As 2* r is the circumference described by the centre of each 
2 ft r 

perforation, — — is the time of a revolution in seconds. 

The quantity —>/~h~Z is also the velocity of the effluent 

water ; therefore, when the machine is working to the greatest 
advantage, the velocity with which water issues is equal to that 
with which it is carried horizontally in an opposite direction ; 
so that, on coming out, it falls perpendicularly down. 

10. The following dimensions have been successfully adopt- 
ed : viz. radius of arms from the centre of pivot to the centre 
of the discharging holes, 46 inches ; inside diameter of the 
arms, 3 inches : diameter of the supplying pipe, 2 inches ; 
height of the working head of water, 21 feet above the points 
of discharge. When the machine was not loaded, and had but 
one orifice open, it made 115 turns in a minute. This agrees 
to a velocity of 46 feet in a second, for the orifice, greater than 
the full velocity due to the head of water by between 9 and 
10 feet : the difference is due to the effect of the centrifugal 
force. 

The theory of this machine is yet imperfect ; but there can 
be no doubt of its utility in cases where the stream is small 
with a considerable fall. 

Mr. James Whiteland, a correspondent of the Franklin 
Journal, proposes to make the horizontal arms of the mill of a 
curved form, such that the water will run from the centre to the 
extremity of the arms in a straight line when the machine is 
working. For the method of constructing the curve, see Me- 
chanic's Magazine, No. 499. It is very clear, however, that 
the additional efficiency of the machine will not be so great by 
any means as the inventor anticipates. 

2f 2 



330 



PNEUMATICS. 



CHAPTER XIII. 
PNEUMATICS. 

Section I. — Equilibrium of air and elastic fluids. 

1. The fundamental propositions that belong to hydrostatics 
are common to the compressible and the incompressible fluids, 
and need not, therefore, be repeated here. 

2. Atmospheric air is the best known of the elastic fluids, 
and has been defined an elastic fluid, having weight, and resist- 
ing compression with forces that are directly as its density, or 
inversely as the spaces within which the same quantity of it is 
contained. 

The correctness of this definition is confirmed by experi- 
ment. 

The weight of air is known from the Torricellian experi- 
ment, or that of the barometer. The air presses on the orifice 
of the inverted tube with a force just equal to the weight of the 
column of mercury sustained in it. 

A bottle, weighed when filled with air, is found heavier than 
after the air is extracted. The pressure of the atmosphere is at 
a mean about 14 lbs. on every square inch of the earth's sur- 
face. Hence the total pressure on the convex surface of the 
earth = 10,686,000,000 hundreds of millions of pounds. 

The elastic force of the air is proved, by simply inverting a 
vessel full of air in water : the resistance it offers to farther 
immersion, and the height to which the water ascends within 
it, in proportion as it is farther immersed, are proofs of the 
elasticity of the air contained in it* 

When air is confined in a bent tube, and loaded with differ- 
ent weights of mercury, the spaces it is compressed into are 

* It is in virtue of this property, and ought to be known as extensively as pos- 
sible, that a man's hat will serve in most cases as a temporary life-preserver, to per- 
sons in hazard of drowning, by attending to the following directions : — When a 
person finds himself in, or about to be in, the water, let him lay hold of his hat 
between his hands, laying the crown close under his chin, and the mouth under the 
water. By this means, the quantity of air contained in the cavity of the hat will be 
sufficient to keep the head above water for several hours, or until assistance can be 
rendered. 



PNEUMATICS : AIR. 331 

found to be inversely as those weights. But those weights are 
the measures of the elasticity ; therefore the elasticities are 
inversely as the spaces which the air occupies. 

The densities are also inversely as those spaces ; therefore 
the elasticity of air is directly as its density. This law was 
first proved by Mariotte's experiments. 

In all this, the temperature is supposed to remain un- 
changed. — These properties seem to be common to all elastic 
fluids. 

Air resists compression equally in all directions. No limit 
can be assigned to the space which a given quantity of air would 
occupy if all compression were removed. 

3. In ascending from the surface of the earth, the density of 
the air necessarily diminishes : for each stratum of air is com- 
pressed only by the weight of those above it ; the upper strata 
are therefore less compressed, and of course less dense than those 
below them. 

4. Supposing the same temperature to be diffused through 
the atmosphere, if the heights from the surface be taken in- 
creasing in arithmetical progression, the densities of the strata 
of air will decrease in geometrical progression. Also, since the 
densities are as the compressing forces, that is, as the columns 
of mercury in the barometer, the heights from the surface being 
taken in arithmetical progression, the columns of mercury in 
the barometer at those heights will decrease in geometrical 
progression. 

As logarithms have, relatively to the numbers which they 
represent, the same property, therefore if b be the column of 
mercury in the barometer at the surface, and /3 at any height h 
above the surface, taking m for a constant coefficient, to be 
determined by experiment, h = m (log b — log j8), or k = 

m log — : where m may be determined by finding trigonome- 

trically the value of h in any case, where b and /3 have been 
already ascertained. 

5. If b be the height of the mercury in the barometer at the 
lowest station, p at the highest, t and /' the temperatures of the 
air at those stations, / the fixed temperature at which no cor- 
rection is required for the temperature of the air ; and if q and 
q' be the temperatures of the quicksilver in the two barome- 
ters, and n the expansion of a column of quicksilver, of which 
the length is 1, for 1° of heat ; h being the perpendicular height 
(in fathoms) of the one station above the other. 



332 PNEUMATICS I ATMOSPHERIC ALTITUDES. 

h = 10000 ( 1 + -00244 / * + *' _ /log.) 



0x(1 + ti( 9 -_ qj 

n being nearly = 

* J 10000 

If the centigrade thermometer is used, because the beginning 
of the scale agrees with the temperature/, so that/= 0, the 
formula is more simple ; and if the expansion for air and mer- 
cury be both adapted to the degrees of this scale, 

h = 10000 (1 + -00441 i t + t\ i 0ff h _ 

V 2 / 6 */3(l + '00018 (q—q') 

6. The temperature of the air diminishes on ascending into 
the atmosphere, both on account of the greater distance from 
the earth, the principal source of its heat, and the greater 
power of absorbing heat that air acquires, by being less com- 
pressed. 

7. Professor Leslie, in the notes on his Elements of Geome- 
try, p. 495, edit. 2d, has given a formula for determining the 
temperature of any stratum of air when the height of the mer- 
cury in the barometer is given. The column of mercury at the 
lower of two stations being b, and at the upper /3, the diminu- 
tion of heat, in degrees of the centigrade, is (- — -=j 25. — 

This seems to agree well with observation. 

8. If the atmosphere were reduced to a body of the same 
density which it has at the surface of the earth, and of the same 
temperature, the height to which it would extend is, in fathoms, 

equal to 4343 (1 + -00441 — — - ), or, taking the expansion 

4 X t\ 
according to Laplace = 4343 (1 -f y^rV 

Hence if b be the height of the mercury in the barometer, 
reduced to the temperature t, the specific gravity of mercury 

4 t \ 
is to that of air, as b to 4343 (1 + tt^;)> or the specific gra- 
vity of air = 



4 t 



12 (4343) (1 + 1000/ 

The divisor 72 is introduced in consequence of b being ex- 
pressed in inches. — [Playfair's Outlines.) 




PNEUMATICS I PUMPS. 333 



Section II. — Pumps. 

1. Def. The term Pump is generally applied to a machine 
for raising water by means of the air's pressure. 

2. The common suction-pump consists of two hollow cylin- 
ders, which have the same axis, and are joined 
in a c. The lower is partly immersed, per- 
pendicularly in a spring or reservoir, and is 
called the suction-tube ; the upper the body 
of the pump. At a c is a fixed sucker con- 
taining a valve which opens upwards, and is 
less than 34 feet from the surface of the water. 
In the body of the pump is a piston d made 
air-tight, moveable by a rod and handle, and 
containing a valve opening upwards. And a 
spout g is placed at a distance greater or less, 
as convenience may require, above the greatest 
elevation of d. 

3. To explain the action of this pump. 

Suppose the moveable piston d at its lowest depression, the 
cylinders free from water, and the air in its natural state. On 
raising this piston, the pressure of the air above it keeping its 
valve closed, the air in the lower cylinder a b forces open the 
valve at a c, and occupies a larger space, viz., between b the 
surface of the water, and d ; its elastic force therefore being 
diminished, and no longer able to sustain the pressure of the 
external air, this latter forces up a portion of the water into the 
cylinder a b to restore the equilibrium. This continues till the 
piston has reached its greatest elevation, when the valve at a c 
closes. In its subsequent descent, the air below d becoming 
condensed, keeps the valve at a c closed, and escapes by forcing 
open that at d, till the piston has reached its greatest depression. 
In the following turns a similar effect is produced, till at length 
the water rising in the cylinder forces open the valve at a c, 
and enters the body of the pump ; when, by the descent of 
d, the valve in a c is kept closed, and the water rises through 
that in d, which on re-ascending, carries it forward, and throws 
it out at the spout g. 

4. Cor. 1. The greatest height to which the water can be 
raised in the common pump by a single sucker is when the 
column is in equilibrio with the weight of the atmosphere, 
that is, between 32 and 36 feet. 

5. Cor. 2. The quantity of water discharged in a given time 
is determined by considering that at each stroke of the piston 

44 



t 



g^E 



334 PNEUMATICS : SUCKING-PUMP. 

a quantity is discharged equal to a cylinder whose base is a sec- 
tion of the pump, and altitude the play of the piston. 

6. To determine the force necessary to over- 
come the resistance experienced by the piston 
in ascending. Let h = the height hf of the 
surface of the water in the body of the pump 
above e f the level of the reservoir ; and a 2 = 
the area of the section m n. Let A'=the height 
of the column of water equivalent to the pres- 
sure of the atmosphere ; and suppose the pis- 
ton in ascending to arrive at any position m n 
which corresponds to the height i f. It is evi- 
dent that the piston is acted upon downwards by 
the pressure of the atmosphere = a 2 A', and by 
the pressure of the column b m = a 2 x hi; 
therefore the whole tendency of the piston to 
descend = a 2 (h' -f- h I.) E F 

But the piston is acted upon upwards by the pressure of the 
air on the external surface e f of the reservoir = a 2 h' ; part 
of which is destroyed by the weight of the column of water 
having for its base m n, and height f i ; 

v the whole action upwards = a 2 x (h' — pi); 

whence f = a 2 , (h' + h i) = a 2 . (h' — f i) 
= a 2 . f h = a 2 h, 
that is, the piston throughout its ascent is opposed by a force 
equal to the weight of a column of water having the same 
base as the piston, and an altitude equal to that of the sur- 
face of the water in the body of the pump above that in the 
reservoir. In order, therefore, to produce the upward motion 
of the piston, a force must be employed equal to that determined 
above, together with the weight of the piston and rod, and the 
resistance which the piston may experience in consequence of 
the friction against the inner surface of the tube.* 

When the piston begins to descend, it will descend by its 
own weight ; the only resistance it meets with being friction, 
and a slight impact against the water. 

7. Cor. 1. If the water has not reached the piston, let its 

* Suppose the body of the pump to be 6 inches in diameter, and the greatest 
height to which the water is raised to be 30 feet ; suppose, also, the weight of the 
piston and its rod to be lOlbs., and the friction one-fifth of the whole weight. Then, 
by the rule at p. 201, JL of the square of the diameter gives the ale gallons in a 
yard in length of the cylinder, and an ale gallon, p. 290, weighs 10y lbs. There- 
fore (6 2 X 10) + JL 6 2 X 10)= 360 + 7-4 = 367-4 lbs. weight of the opposing 
column of water. And 367-4-f-10-f-j (377-4)=452-9 lbs., whole opposing pres- 
sure. 

If the piston rod be moved by a lever whose arms are as 10 to 1, this pressure 
will be balanced by a force of 45-29 lbs., and overcome by any greater force. 



PNEUMATICS I SUCKING-PUMP. 



335 



k 



level be in v z. The under surface of the piston will be 
pressed by the internal rarefied air. But this air, together with 
the column of water, e v, is in equilibrio with the pressure of 
the atmosphere a 3 h' ; and .*. its pressure=a 2 . (h' — e v). And 
the pressure downwards = a 2 h' ; 

.♦. p=fl 2 XE v. 
Hence the force requisite to keep the piston in equilibrio in- 
creases as the water rises and becomes constant, and = a 2 h as 
soon as the water reaches the constant level b h. 

8. Cor. 2. If the weight of the piston be taken into the ac- 
count, let this weight be equal to that of a column of water 
whose base is m n and height p, =a 2 p ; 

,\ f = a 2 . (e v + p). 

9. To determine the height to which the water will rise after 
one motion of the piston ; the fixed sucker being placed at the 
junction of the suction-tube and body of the 
pump : supposing that after every elevation of the 
piston there is an equilibrium between the pres- 
sure of the atmosphere on the surface of the water 
in the reservoir, and the elastic force of the rare- 
fied air between the piston and surface of the 
column of water in the tube, together with the 
weight of that column. 

Let a b be the surface of the water in the suc- 
tion-tube, after the first stroke of the piston : if 
the piston were for an instant stationary at d, 
the pressure of the atmosphere would balance e b, 
and the elastic force of the air in n a. 

Let A e the height of the suction-tube = a, 
d r the play of the piston = b, 
h — the height of a column of water equivalent to the pres- 
sure of the atmosphere, 
y — the height of a column equivalent to the pressure of the 

air in n a, 
x = e a, 

and r and r — the radii of the body and the suction- 
tube. 



EC 



Then x+y=h, 



rt r 2 a 



and y = h. — = h . — - , 

* n a n r 3 b + rt r 2 . (a — x) 

hr* a 



whence h = x + 



r 2 b + r z a — r* x 
hr* a 



r 2 6 + r 2 a—r % x ' 



336 PNEUMATICS : SUCKING-PUMP. 

.-. /* r 2 b+h r* a — h r 2 ^=r 2 b x+r* a x — r* x*+h r* a, 

R 3 R 3 

or a: 3 — (h + -5 • b + a) . x = — h . — 2 . b, 
and x 2 — p x = — h m b, 

R 2 

if m be = -5, and p=^ + w& + «; 
.\# = § . lp± \/> 3 — 4 hm b}, 



and y = § . J2 A — p =F y/p 2 —4 h m b\, 

only one of which values will be applicable, viz. that which an- 
swers to the lower sign ; since x and y must be less than h ; 
and if the upper sign be used, x will be found greater than h. 

10. Having given the height of the water raised, and that due 
to the pressure of air in the pump after the first ascent of the 
piston ; to determine them for the second, third, &c. ascents. 

Let e a' represent the height of the water after the second 
ascent, and let it = x x , 

and let y x = the height due to the elastic force of the air ; 

then x x -f- y x = h ; 
and y± = y . — „ since the air which occupied c a now occu- 
pies n a'; 



. v = yr*.(a — x) _ y . (a—x) 
" yx R 2 b + r 2 .(a — x x ) mb+a — x x 

whence h = x x + ,*."" , 
mb-\-a — x x 



and .*. x x = h . {p — Vp 2 — 4 h m b — 4 x . (p -\- a — x)], 
and y x = % . {2 h—p + y/p 2 ~—4hmb—4 x . (h+a — x)}. 
From these are deduced values of x z , y 2 , x 3 , y 3 , &c. 
x 2 — 5 . \p — </p Q — 4 hm b — 4 x x . h + a — x t ) }, 



y % -=z \ . \2 h—p + s/p* — 4 hmb — 4 x x . (h+a — x x )}, 

and so on. Whence if x n be taken to represent the height of 
the water after (n + 1) ascents, 

x n — 5 . \p — \/p* — 4 hmb— 4 x n _ x . (h + a — x n _ t )], 

andy n =5. \2h— p+ <Sf—4 h m 6—4 x n _ x . (h + a— x n - x )}. 



PNEUMATICS ! STJCKING-PTJMP. 337 

11. Cor. 1. Hence may be determined the height to which 
the water can rise after any given number of ascents of the pis- 
ton, and the elastic force of the air in the suction-tube. 

12. Cor. 2. Knowing the elevation due to each particular 
stroke, the differences of those elevations, and the successive 
differences in the elastic force of the remaining air, may be 
known. 

1 3. Cor. 3. If the weight of the valve c be not considered, it 
is evident that after a certain number of strokes a vacuum will 
be produced in the suction-tube, provided it be equal to, or not 
greater than the height due to the pressure of the atmosphere, 
that is, if a be not greater than h. 

For in this case, x n — x n _ iy 

and .*. #«_!=£ \p — s/fi 2, — 4 hm b — 4a? n _ 1 . (A + tf — a?«_i) }, 
whence a; a - 1 = A, the greatest height of the column of water in 
the tube. If therefore the length of the suction-tube do not ex- 
ceed the height due to the pressure of the atmosphere, the water 
will continue to ascend in it after every stroke of the piston, till 
at length it will pass into the body of the pump. 

But if the altitude of a f be greater than A, the water will 
continue to ascend without ever reaching its maximum height. 
For in this case, an actual vacuum cannot be produced ; and as 
x n +y n =h, and y n can never become=o ; .*. x n can never = h.* 
But the successive values of y continually decreasing, the cor- 
responding values of x will continually increase. 

14. Cor. 4. If the weight of the valve c be taken into the ac- 
count, a column of water must be added equal to the additional 
pressure to be overcome. Let / = the height of this column, 
then 

x + y + / = h; 
and .*. x + y = h — / = h. 

If therefore this value of A' be substituted for h, the preceding 
equations are applicable. 

15. Cor. 5. In the preceding cases, the moveable piston has 
been supposed to descend to a c. If it does not, it may happen 
that the water may not reach a c, though a c be less than 34 
feet from the surface of the water in the reservoir. 

After the first elevation of the moveable piston to its 
greatest altitude, c being closed, the elastic force of the air 
between d n and a c is (A — x), and its magnitude * b r 3 . If 



* Hence it appears that it is not strictly true, that water will ascend in the suc- 
tion-tube to a height equal to that of a column equivalent to the pressure of the 
atmosphere. This is a limit to which it approximates, but does not reach in a 
finite time. 

2G 



338 PNEUMATICS I SUCKING-PUMP. 

in descending, the piston describes a space b' less than b, so as 
to stop at a distance b — b' from a c, this magnitude becomes 

(b — b') . n b 3 ; .*. the elastic force is (h — x) . -7 r> Now 

in order that the pressure upwards may open the valve, this 
must exceed the elastic force of the atmosphere ; 

... (h — x).£—-^>h, 
or(h — x).b>h.(b — V); 

Z. ill x ^ h ' 

.'. b x < h b\ or -t-<-t-. 
ft o 

b' x 

If .-. t- be less than -7-, the valve d n will not open ; 

there will therefore be the same quantity of air between A c 
and the sucker : which, when the piston has reached its highest 
elevation, will have the same elastic force as that between a c 
and a' b' ; and therefore c being equally pressed on both sides, 
will remain unmoved, and the water will not ascend. 

1 6. If the fixed sucker be placed at the surface of the water ; 
to determine the ascent of the water in the suction-tube. 

Let e «, e a' be the successive heights to which the water 
rises ; then after the first ascent of the piston, 

x + y = hy 

j ha 

and y= — j— : 

* mb-\-a — x 



whence x = \ . { p — s/p 2 — 4 h m b] 

and y = \ . {2 h — p + y/p*— 4 hmb], 

which equations are the same as were determined for the first 
ascent of the piston (9). Therefore, in the same manner as 
before, 



we shall have x n = h . {p — <Sp* — 4hm b—4hx n _ ± }y 
y n = \ . {2 h—p+ y/p* — 4 h m b — 4 h a?._J. 

17. Cor. 1. If the water be supposed to stop after (n + 1) 
ascents of the piston, then x n = x u - x ; 



and .*. av-i = §.{/> — </p 2 — 4 hmb — 4 h x n _J, 



whence x n _ x = h • \a+m b ± s/(a + m b 2 — 4hmb\. 



PNEUMATICS : SUCKING-PUMP. 339 

Hence, therefore, there are two altitudes at which the water 
may stop in its ascent, if (a + m b) 2 is equal to or greater than 
4 h m b. In the former case the two values of x n _ ± are equal, 
that is, there will be only one altitude = § . (a -f- m b), at which 
the water will stop. In the latter case there are two which 
may be ascertained. 

If 4 h m b be greater than (a + m b) z , the water will not 
stop. 

Ex. 1. If h = 32 feet, a = 20, b == 4, and m = 1, or the 
suction-tube and body of the pump be of th e same diameter, 
^„_!=5. {20 -f 4± V (24) 3 — 4.1.32.4} = §. f 24±n/64] 
= 16 or 8. 

,£r. 2. If h = 32 feet, a = 25, 6 = 2, and m = 4, 
aV^ i . f25 + 8 ± v/ (33) a — 4.32.4.2} = § . J33 ± 
\/65} = 41-8062 or 24*1938. 

18. Cor. 2. If w = 1, or the tubes have the same diameter, 

av-* = i . {a +♦£ ± x/ (a + 6) 2 — 4Ao }, 
which is imaginary, if (a + 6) 3 is less than 4 h b, or b greater 
- (a + J)' 
than 4 A ' 

In order, therefore, that this pump may produce its effect, 
the play of the piston must be greater than the square of its 
greatest altitude above the surface of the water in the reser- 
voir divided by four times the height due to the pressure of 
the atmosphere. 

19. The lifting-pump consists of a hollow cylinder, the 
body of which is immersed in the reservoir. It is furnished 
with a movable piston, which, entering below, lifts the water 
up, and is movable by means of a frame which is made to ascend 
and descend by a handle. The piston is furnished with a valve 
opening upwards. A little below the surface of the water 
is a fixed sucker with a valve opening upwards. This is an 
inconvenient construction, upon the peculiarities of which we 
need not dwell. 



340 



PNEUMATICS I FORCING-PUMP. 




20. The forcing-pump consists of a 
suction-tube a e f c, partly immersed in 
the reservoir, and of the body of the pump 
a b g c, and of the ascending tube h m. 
The body is furnished with a movable solid 
sucker or plunger d, made air-tight. And 
at a c and h are fixed suckers with valves 
opening upwards. 

21. To explain the action of this pump. 
Suppose the plunger d at its greatest de- 
pression ; the valves closed, and the air in 
its natural state. Upon the ascent of d, 
the air in a c d occupying a greater space, 
its elasticity will be diminished, and consequently the greater 
elasticity of the air in a f will open the valve at a c, whilst that 
at h is kept closed by the elasticity of the external air ; water 
therefore will rise in the suction-tube. On the descent of d 
from its greatest elevation, the elasticity of the air in the body 
of the pump will keep the valve a c closed, and open that at h, 
whence air will escape. By subsequent ascents of the piston, 
the air will be expelled, and water rise into the body. The 
descending piston will then press the water through the valve 
at h, which will close, and prevent its return into the body of 
the pump ; d therefore ascending again, the space left void will 
be filled by water pressing through the valve a c ; and this 
upon the next ascent of d is forced into the ascending tube ; 
and thus by the ascents and descents of d, water may be raised 
to the required height. 

22. Cor. In this pump d must not ascend higher than about 
32 feet from the surface of the water in the reservoir. 

23. To determine the force necessary to overcome the resist- 
ance experienced by the piston. 

Let h — the height of a column of water equivalent to the 
pressure of the atmosphere, and eb the height to which the 
water is forced. Let m n be any position of the piston d whose 
area = a, and the weight of the piston and its appendages = p. 
Let x = the force necessary to push the piston upwards during 
the suction, friction not being considered, and y = that em- 
ployed to force it down. 

When the piston ascends, and h is closed 

x = p + aA — a . [h — me) 

=a P + A . M E. 

Let the sucker be in the same position in its descent, and 
therefore a c closed, and h open. 



PNEUMATICS : FORCING-PUMP. 



341 



y = aA + a.mb — (a h + f) 

= A . M B P. 

Hence x + y = a . e b ; or the whole force exerted in the 
case of equilibrium is equal to the weight of a column of 
water whose base is equal to that of the piston, and altitude the 
distance between the surface of the water and the point to which 
it is to be raised. 

24. Cor. 1. In this pump the effort is divided into two parts, 
one opposed to the suction, and the other to the forcing ; whereby 
an advantage is gained over the other pumps where the whole 
force is exerted at once whilst the water is raised. 

25. Cor. 2. In order to have the force applied uniform, let 
x = y; 

.*. p + a.me=a.mb — p; 
.*. p = h a . (m b — me). 

The piston therefore must play in such a manner that m b may 
be greater than m e. 

26. Cor. 3. In the common forcing- 
pump the stream is intermitting ; for 
there is no force impelling it during the 
return of the sucker. 

One mode of remedying this, is by 
making an interruption in the ascending 
tube, which is surrounded by an air-ves- 
sel t ; in which when the water has risen above z, the air above 
it is compressed, and by its elasticity forces the water up through 
z ; the orifice of which is narrower than that of the tube, and 
therefore the quantity of water introduced during the descent 
of the piston will supply its discharge for the whole time of the 
stroke, producing a continued stream. 

27. The fire-engine consists of a large receiver abcd, 
called the air-vessel, into which 
water is driven by two forcing- 
pumps, e f, g h, (whose pistons 
are q and r), communicating 
with its lower extremities at i 
and k, through two valves open- 
ing inwards. From the receiver 
proceeds a tube m l through 
which the water is thrown, and 
directed to any point by means of a pipe moveable about the 
extremity l. The pumps are worked by a lever, so that 

45 2 g 2 





342 pneumatics: fire-engine. 

whilst one piston descends the other ascends. The pumps com- 
municate with a reservoir of water at n. 

28. To explain the action of this engine. 

The tube n being immersed in the reservoir, and the piston r 
drawn up, the pump g h becomes filled ; and the descent of the 
piston r will, as in the forcing-pump (21), keep the valve h 
close, and cause the water to pass into the air-vessel by the 
valve i, whilst by the weight of the water in the air-vessel, the 
valve k will be kept shut. In the same manner, when r ascends, 
q descending will force the water through k into the air-vessel. 
By this means the air above the surface of the water becoming 
greatly compressed, will by its elasticity force the water to as- 
cend through m l, and to issue with a great velocity from the 
pipe.* 

29. When the air-vessel is half full of water, the air being 
then compressed into half its natural space, will have an elastic 
force equal to twice the pressure of the atmosphere : therefore, 
when the stop-cock is turned, the air within pressing on the 
subjacent water with twice the force of the external air, will 
cause the water to spout from the engine to the height of 
(2 — 1) 33, or 33 feet ; except so far as it is diminished by 
friction. 

n i 

Or, generally, if denote the fractional height of the 

Tt 

water in the air barrel, then — will denote the height of the 

space occupied by the compressed air n times the pressure of 
the atmosphere its elastic force, and (n — 1) 33, the height in 
feet to which the water may be projected. 

Thus, if 1 of the air barrel be the height of the water, the 
elastic force of the air will be four times the pressure of the 
atmosphere, and (4 — 1) 33 = 99 feet, the height to which the 
water may then be thrown by the engine. 

30. The modifications in the constructions of pumps with a 
view to their practical applications are very numerous. Those 
who wish to acquaint themselves with some of the most useful, 
may consult the 2d vol. of my Treatise on Mechanics, and 
Nos. 13, 41, 69, and 93, of the Mechanics 7 Magazine. 

In addition to these, there may now be presented a short 

* The preceding part of this section is taken from Bland? s Hydrostatics ; a very 
elegant and valuable work, which I beg most cordially to recommend to those who 
wish to obtain a comorehensive knowledge of the theory of this department of 
science. 



PNEUMATICS I QUICKSILVER PUMP. 



343 




account of a quicksilver pu?np, which has been recently in- 
vented by Mr. Thomas Clark of Edinburgh, and which works 
almost without friction. It has great power in drawing and 
forcing water to any height, and is extremely simple in its 
construction. 

a a is the main pipe inserted into the 
well b ; a valve is situated at c, and 
another at d, both opening upwards ; 
a piece of iron tube is then bent into a 
circular form, as at f, again turned off 
at g in an angular direction, so as to 
pass through a stuffing box at h, and 
from thence bent outwards as at i, con- 
necting itself with the ring. A quan- 
tity of quicksilver is then put into the 
ring filling it from q to q, and the ring 
being made to vibrate upon its axis h, 
a vacuum is soon effected in the main 
pipe by the recession of the mercury 
from g to q, thereby causing the water to rise and fill the 
vacuum : upon the motion being reversed, the quicksilver 
slides back to g, forces up the water and expels it at the 
spout e. 

" Mr. Clark calculates that a pump of this description with 
a ring twelve feet in diameter, will raise water the same height 
as the common lifting pump, and force it one hundred and fifty 
feet higher without any friction." (Mechanics' Register, and 
Jamieson's Edinburgh Journal.) 

31. It is usual to class with pumps, the machine known by 
the name of Archimedes' screw or the water-snail. This con- 
sists either of a pipe wound spirally round a cylinder, or of one 
or more spiral excavations formed by means of spiral projec- 
tions from an internal cylinder, covered by an external cylin- 
drical case, so as to be water tight. -The cylinder which carries 
the spiral is placed aslant, so as to be inclined to the horizon in 
an angle of from 30° to 45°, and capable of turning upon pivots 
in the direction of its axis posited at each extremity. The 
lower end of the spiral canal being immersed in the river or 
reservoir from which water is to be raised, the water descends 
at first in the said canal solely by its gravity ; but the cylinder 
being turned, by human or other energy, the water moves on 
in the canal, and at length it issues at the upper extremity of 
the tube. 

Several circumstances tend to make this instrument imper- 
fect and inefficacious in its operation. The adjustments 




344 PNEUMATICS : SPIRAL PUMP. 

necessary to ensure a maximum of effectual work are often 
difficult to accomplish. It seldom happens, therefore, that 
the measure of the work done exceeds a third of the power 
employed : so that this apparatus, notwithstanding its apparent 
ingenuity and simplicity, is very sparingly introduced by our 
civil engineers. 

32. Spiral Pump. This machine is formed by a spiral pipe 
of several convolutions, arranged either in a single plane, as 
in the marginal diagram, or upon a 
cylindrical or conical surface, and re- 
volving round an axis. The curved 
pipe is connected at its inner end, 
by a central water-tight joint, to an 
ascending pipe, r p, while the other 
end, s, receives, during each revolu- 
tion, nearly equal quantities of air 
and water. This apparatus is usually 
called the Zurich machine, because it 
was invented, about 1746, by Andrew 

Wirtz, an inhabitant of Zurich. It has been employed with 
great success at Florence, and in Russia ; and the late Dr. 
Thomas Young states, that he employed it advantageously for 
raising water to a height of forty feet. The outer end of the 
pipe is furnished with a spoon s, which contains as much water 
as will half fill one of its coils. The water enters the pipe a 
little before the spoon has reached its highest position, the 
other half remaining full of air. This air communicates the 
pressure of the column of water to the preceding portion ; and 
in this manner the effect of nearly all the water in the wheel 
is united, and becomes capable of supporting the column of 
water, or of water mixed with air, in the ascending pipe. The 
air nearest the joint is compressed into a space much smaller 
than that which it occupied at its entrance ; so that, when the 
height is considerable, it becomes advisable to admit a larger 
portion of air than would naturally fill half the coil. This, 
while it lessens the quantity of water raised, lessens also the 
force requisite to turn the machine. The loss of power, sup- 
posing the machine well constructed, arises only from the 
friction of water on the pipes, and that of the wheel on its 
axis : and where a large quantity is to be raised to a moderate 
height, both of these sources of resistance may be rendered 
very inconsiderable. 

33. Schemnitz Vessels, or the Hungarian Machine. The 
mediation of a portion of air is employed for raising water, 
not only in the spiral pump, but also in the air vessels of 



PNEUMATICS I SCHEMNITZ VESSELS. 



345 



Schemnitz, as shown in the annexed dia- 
gram. A column of water, descending 
through a pipe, c, into a closed reservoir, 
b, containing air, obliges the air to act, by- 
means of a pipe, d, leading from the upper 
part of the air vessel, or reservoir, on the 
water in a second reservoir, a, at any dis- 
tance either above it or below it, and forces 
this water to ascend through a third pipe, 
e, to any height, less than that of the first 
column. The air vessel is then emptied, 
the second reservoir filled, and the whole 
operation repeated. The air, however, 
must acquire a density equivalent to the 
requisite pressure before it can begin to 
act : so that, if the height of the columns 
were 34 feet, it must be reduced to half 
before any water could be raised, and thus 
would be lost. 




its natural space 
half of the force 
But, where the height is small, the height lost 
in this manner is not greater than what is usually spent in over- 
coming friction, and other imperfections of the machinery em- 
ployed. The force of the tide, or of a river rising and falling 
with the tide, might easily be applied to the purpose of raising 
water by a machine of this kind. Thus, if at low tide the ves- 
sel a were filled with air ; then, at high tide, the water flowing 
down the tube e, would cause the water in the vessel b to as- 
cend in the pipe c. 

34. The Hydraulic Ram. In this hydraulic arrangement, 
the momentum of a stream of water flowing through a long 
pipe is employed to raise a small quantity of water to a con- 
siderable height. The passage of the pipe being stopped by 
a valve which is raised by the stream, as soon as its motion 
becomes sufficiently rapid, the whole column of fluid must ne- 
cessarily concentrate its action almost instantaneously upon the 
valve. In these circumstances it may be regarded as losing 
the characteristic property of hydraulic pressure, and to act 
almost as though it were a single solid : so that, supposing the 
pipe to be perfectly elastic and inextensible, the impulse may 
overcome almost any pressure that may be opposed to it. If 
another valve opens into a pipe leading to an air vessel, a 
certain quantity of the water will be forced in, so as to con- 
dense the air, more or less rapidly, to the degree that may be 
required for raising a portion of the water contained in it to a 
given height. 

The late Mr. Whitehurst appears to have been the first 
who employed this method : it was afterwards improved by 



346 



PNEUMATICS : HYDRAULIC RAM. 



Mr. Boulton. But, like many English inventions, it never 
was adequately estimated, until it was brought into public no- 
tice by a Frenchman. M. Montgolfier, its re-inventor, gave to 
it the name which it now bears of the Hydraulic Bam, in al- 
lusion to the battering ram. 

The essential parts of this machine are represented in the 
annexed diagram. When the water in the pipe a b (moving 




in the direction of the arrows) has acquired sufficient velocity, 
it raises the valve b, which immediately stops its farther pas- 
sage. The momentum which the water has acquired then 
forces a portion of it through the valve, c, into the air vessel, d. 
The condensed air in the upper part of d causes the water to 
rise into the pipe e, as long as the effect of the horizontal 
column continues. When the water becomes quiescent, the 
valve b will open again by its own weight, and the current 
along a b will be renewed, until it acquires force enough to 
shut the said valve b, open c, and repeat the operation. 

The motion in the horizontal tube arises from the accelera- 
tion of the velocity of a liquid mass falling down another tube, 
and communicating with this. 

In an experiment made upon an hydraulic ram, at Avilly, 
near Senlis, by M. Turquet, bleacher, the expense of power was 
found to be to the produce, as 100 to 62. In another, as 100 
to 55 : in two others, as 100 to 57. So that a hydraulic ram, 
placed not in unfavourable circumstances, may be reckoned to 
employ usefully rather more than half its force. 

%* For more full accounts of the three last contrivances, the 
reader may consult the 2d volume of my Mechanics. 



Section III. — Wind and Windmills. 

1. Air, when in continuous motion in one direction, be- 
comes a very useful agent of machinery, of greater or less 
energy, according to the velocity with which it moves. Were 



PNEUMATICS : WIND AND WINDMILLS. 



347 



it not for its variability in direction and force, and the conse- 
quent fluctuations in its supply, scarcely any more appropriate 
first mover could generally be wished for. And even with all 
its irregularity, it is still so useful as to require a separate con- 
sideration. 

2. The force with which air strikes against a moving sur- 
face, or with which the wind strikes against a quiescent sur- 
face, is nearly as the square of the velocity : or, more cor- 
rectly, the exponent of the velocity determined according to 
the rule given at pa. 103, varies between 2*03 and 2*05 ; so 
that in most practical cases, the exponent 2, or that of the 
square, may be employed without fear of error. If I be the 
angle of incidence, s 2 the surface struck in feet, and v the 
velocity of the wind, in feet, per second ; then for the force in 
avoirdupois pounds, either of the two following approximations 

, , . n v 2 s z sin 3 1 
may be used : viz./ = — — 

or /= -002288 v 2 s* sin 3 1. 

Of these, the first is usually the easiest in operation, requir- 
ing only two lines of short division, viz. by 40 and by 11. 

If the incidence be perpendicular, sin 2 1 = 1, and these be- 
come, 

/= J^-= -002288 v 2 s 2 . 
J 440 



3. The table in the margin exhi- 
bits the force of the wind when blow- 
ing perpendicularly upon a surface of 
one foot square, at the several velocities 
announced. The velocity of 80 miles 
per hour is that by which the aeronaut 
Garnerin was carried in his balloon 
from Ranelagh to Colchester, in June, 
1802. It was a strong and boisterous 
wind ; but did not'assume the character 
of a hurricane, although a wind with 
that velocity is so characterized in 
Rouse's table. In Mr. Green's aerial 
voyage from Leeds, in September, 1823, 
he travelled 43 miles in 18 minutes, 
although his balloon rose to the height 
of more than 4000 yards. 



Borda found by experiment in the year 1762, that the force 
of the wind is very nearly as the square of the velocity, but 



Velocity of the 
Wind. 


Perpendi- 
cular force 






on one sq. 
foot in a- 
voirdupois 
pounds. 


Miles 
in one 
hour 


= feet 
in one 
second 


1 


1-47 


•005 


2 


2-93 


•020 


3 


4-40 


•044 


4 


5-87 


•079 


5 


7-33 


•123 


10 


14-67 


•492 


15 


22-00 


1-107 


20 


29-34 


1-968 


25 


36-67 


3-075 


30 


44-01 


4-429 


35 


51-34 


6-027 


40 


58-68 


7-873 


45 


66-01 


9-963 


50 


73-35 


12-300 


60 


88-02 


17-715 


80 


117-36 


31-490 


100 


146-70 


49-200 



348 PNEUMATICS : WIND AND WINDMILLS. 

he assigned that force to be greater than what Rouse found 
(as expressed in the above table) in the ratio of 111 to 100. 
Borda ascertained also, as was natural to expect, that, upon 
different surfaces with the same velocity, the force increased 
more rapidly than the surface. M. Valtz, applying the me- 
thod of the minimum squares to Borda's results, ascertained 
that the whole might be represented by the formula 

y = 0-001289 x 2 +0-000050541 x* 
and nearly as correctly by y = 0*00108 x 2 -* 

x* representing the surface in square inches (French), and y 
the force corresponding to the velocity of 10 feet per second 
expressed in French pounds. 

4. In the application of wind to mills, whatever varieties 
there may be in their internal structure, there are certain 
rules and maxims with regard to the position, shape, and 
magnitude of the sails, which will bring them into the best 
state for the action of the wind, and the production of use- 
ful effect. These have been considered much at large by 
Mr. Smeaton : for this purpose he constructed a machine, 
of which a particular description is given in the Philosophi- 
cal Transactions, vol. 51. By means of a determinate weight 
it carried round an axis with an horizontal arm, upon which 
were four small moveable sails. Thus the sails met with a 
constant and equable blast of air ; and as they moved round, 
a string with a weight affixed to it was wound about their 
axis, and thus showed what kind of size or construction of 
sails answered the purpose best. With this machine a great 
number of experiments were made ; the results of which were 
as follow : 

(1.) The sails set at the angle with the axis, proposed as the 
best by M. Parent and others, viz. 55°, was found to be the 
worst proportion of any that was tried. 

(2.) When the angle of the sails with the axis was increased 
from 72° to 75°, the power was augmented in the proportion of 
31 to 45 ; and this is the angle most commonly in use when the 
sails are planes. 

(3.) Were nothing more requisite than to cause the sails 
to acquire a certain degree of velocity by the wind, the po- 
sition recommended by M. Parent would be the best. But 
if the sails are. intended with given dimensions to produce 
the greatest effects possible in a given time, we must, if planes 
are made use of, confine our angle within the limits of 72 and 
75 degrees. 

(4.) The variation of a degree or two, when the angle is 
near the best, is but of little consequence. 



PNEUMATICS I WIND AND WINDMILLS. 349 

(5.) When the wind falls upon concave sails, it is an advan- 
tage to the power of the whole, though each part separately 
taken should not be disposed of to the best advantage. 

(6.) From several experiments on a large scale, Mr. Smea- 
ton has found the following angles to answer as well as any. 
The radius is supposed to be divided into six parts ; and |th, 
reckoning from the centre, is called i, the extremity being 
denoted 6. 

Angle with Angle with 

No « that axis. the pane of 

motion. 

1 72° 18° 

2 71 19 

3 . . 72 18 middle 

4 . 74 16 

5 77£ I2i 

6 83 7 extremity. 

(7.) Having thus obtained the best method of weathering 
the sails, i. e. the most advantageous manner in which they can 
be placed, our author's next care was to try what advantage 
could be derived from an increase of surface upon the same 
radius. The result was, that a broader sail requires a larger 
angle ; and when the sail is broader at the extremity than near 
the centre, the figure is more advantageous than that of a 
parallelogram. The figure and proportion of enlarged sails, 
which our author determines to be most advantageous on a 
large scale, is that where the extreme bar is one-third of the 
radius or whip (as the workmen call it), and is divided by the 
whip in the proportion of 3 to 5. The triangular or loading 
sail is covered with board from the point downward of its 
height, the rest as usual with cloth. The angles above men- 
tioned are likewise the most proper for enlarged sails; it 
being found in practice that the sails should rather be too 
little than too much exposed to the direct action of the 
wind. 

Some have imagined, that the more sail the greater would 
be the power of the windmill, and have therefore proposed to 
fill up the whole area ; and by making each sail a sector of an 
ellipsis, according to M. Parent's method, to intercept the 
whole cylinder of wind, in order to produce the greatest effect 
possible. From our author's experiments, however, it appeared, 
that when the surface of all the sails exceeded seven-eighths of 
the area, the effect was rather diminished than augmented. 
Hence he concludes, that when the whole cylinder of wind is 
intercepted, it cannot then produce the greatest effect for want 
of proper interstices to escape. 

46 2H 



350 PNEUMATICS : WIND AND WINDMILLS. 

" It is certainly desirable (says Mr. Smeaton), that the sails 
of windmills should be as short as possible ; but it is equally 
desirable, that the quantity of cloth should be the least that may 
be, to avoid damage by sudden squalls of wind. The best 
structure, therefore, for large mills, is that where the quantity 
of cloth is the greatest in a given circle that can be : on this 
condition, that the effect holds out in proportion to the quantity 
of cloth ; for otherwise the effect can be augmented in a given 
degree by a lesser increase of cloth upon a larger radius than 
would be required if the cloth was increased upon the same 
radius." 

(8.) The ratios between the velocities of windmill sails un- 
loaded, and when loaded to their maximum, turned out very 
different in different experiments ; but the most common pro- 
portion was as 3 to 2. In general it happened that where the 
power was greatest, whether by an enlargement of the surface 
of the sails, or an increased velocity of the wind^ the second 
term of the ratio was diminished. 

(9.) The ratios between the least load that would stop the 
sails and the maximum with which they would turn, were 
confined betwixt that of 10 to 8 and 10 to 9 ; being at a me- 
dium about 10 to 8*3, and 10 to 9, or about 6 to 5 ; though on 
the whole it appeared, that where the angle of the sails or 
quantity of cloth was greatest, the second term of the ratio was 
less. 

(10.) The velocity of windmill sails, whether unloaded or 
loaded, so as to produce a maximum, is nearly as the velocity 
of the wind, their shape and position being the same. On this 
subject Mr. Ferguson remarks, that it is almost incredible to 
think with what velocity the tips of the sails move when acted 
upon by a moderate wind. He has several times counted the 
number of revolutions made by the sails in 10 or 15 minutes ; 
and, from the length of the arms from tip to tip, has com- 
puted, that if an hoop of the same size was to run upon plain 
ground with an equal velocity, it would go upwards of 30 miles 
in an hour. 

(11.) The load at the maximum is nearly, but somewhat less 
than, as the square of the velocity of the wind ; the shape and 
position of the sails being the same. 

(12.) The effects of the same sails at a maximum are nearly, 
but somewhat less than, as the cubes of the velocity of the 
wind. 

(13.) The load of the same sails at a maximum is nearly as 
the squares, and the effect as the cubes of their number of turns 
in a given time. 

(14.) When sails are loaded so as to produce a maximum 



351 

at a given velocity, and the velocity of the wind increases, the 
load continuing the same ; then the increase of effect, when 
the increase of the velocity of the wind is small, will be nearly 
as the squares of these velocities : but when the velocity of the 
wind is double, the effects will be nearly as 10 to 21 h ; and 
when the velocities compared are more than double of that 
where the given load produces a maximum, the effects increase 
nearly in a simple ratio of the velocity of the wind. Hence 
our author concludes that windmills, such as the different spe- 
cies for draining water, &c. lose much of their effect by acting 
against one invariable opposition. 

(15.) In sails of a similar figure and position, the number of 
turns in a given time will be reciprocally as the radius or length 
of the sail. 

(16.) The load at a maximum that sails of a similar figure 
and position will overcome, at a given distance from the centre 
of motion, will be as the cube of the radius. 

(17.) The effects of sails of similar position and figure are 
as the square of the radius. Hence augmenting the length of 
the sail without augmenting the quantity of cloth, does not in- 
crease the power ; because what is gained by length of the lever 
is lost by the slowness of the motion. Hence also, if the sails 
are increased in length, the breadth remaining the same, the ef- 
fect will be as the radius. 

(18.) The velocity of the extremities of the Dutch sails, as 
well as of the enlarged sails, either unloaded or even when 
loaded to a maximum, is considerably greater than that of the 
wind itself. This appears plainly from the observations of 
Mr. Ferguson, already related, concerning the velocity of 
sails. 

(19.) From many observations of the comparative effects of 
sails of various kinds, Mr. Smeaton concludes, that the en- 
larged sails are superior to those of the Dutch construction. 

(20.) He also makes several just remarks upon those wind- 
mills which are acted upon by the direct impulse of the wind 
against sails fixed to a vertical shaft : his objections have, we 
believe, been justified in every instance by the inferior efficacy 
of these horizontal mills. 

" The disadvantage of horizontal windmills (says he) does 
not consist in this, that each sail, when directly opposed to 
the wind, is capable of a less power than an oblique one of 
the same dimensions ; but that in an horizontal windmill little 
more than one sail can be acting at once : whereas in the com- 
mon windmill, all the four act together ; and therefore, sup- 
posing each vane of an horizontal windmill to be of the same 
size with that of a vertical one, it is manifest that the power of 



352 pneumatics : smeaton's rules for windmills. 

a vertical mill will be four times as great as that of an horizon- 
tal one, let the number of vanes be what they will. This dis- 
advantage arises from the nature of the thing ; but if we consider 
the further disadvantage that arises from the difficulty of getting 
the sails back again against the wind, &c. we need not wonder 
if this kind of mill is in reality found to have not above one- 
eighth or one-tenth of the power of the common sort ; as has 
appeared in some attempts of this kind." 



Coulomb's Experiments. 

5. M. Coulomb, whose experiments have tended to the elu- 
cidation of many parts of practical mechanics, devoted some 
time to the subject of windmills. The results of his labours 
were published in the Memoirs of the Paris Academy for 1781. 
The mills to which he directed his attention were in the vi- 
cinity of Lille, and were, in fact, oil mills. From the outer 
extremity of one sail to the corresponding extremity of the 
opposite sail, was 70 feet, the breadth of each sail 6i feet, of 
which the sail-cloth when extended occupies 5§ feet, being 
attached on one side to a very light plank ; the line of junc- 
tion of the plank and of the sail-cloth, forms, on the side struck 
by the wind, an angle sensibly concave at the commencement 
of the sail, but diminishes gradually all along so as to vanish at 
the remoter extremity. The angle with the axis, at seven feet 
from the shaft, is 60°, and it increases continually, so as to 
amount to nearly 84° at the extremity. The shaft upon which 
the sails turn, is inclined to the horizon, in different angles in 
different mills, from 8 to 15 degrees. 

Coulomb infers from his experiments, 

(1.) That the ratio between the space described by the wind 
in a second, and the number of turns of a sail in a minute, is 
nearly constant, whatever be the velocity of the wind ; the said 
ratio being about 10 to 6, or 5 to 3. 

(2.) That with a wind whose velocity is 21 § feet (English) 
per second, the quantity of action produced by the impulsion 
of the wind is equivalent to a weight of 1080 pounds avoirdu- 
pois raised 270 feet in a minute ; the useful effect being equiva- 
lent to a weight of 1080 pounds raised 232 feet in the same 
time : whence it results that the quantity of effect absorbed by 
the stroke of the stampers, the friction, &c. is nearly a sixth 
part of the quantity of action. 

(3.) Suppose one of these mills to work 8 hours in a day. 
Coulomb regards its daily useful effect as equivalent to that 



PNEUMATICS : WINDMILLS. 353 

of 1 1 horses working at a walking-wheel, in a path of the usual 
radius. 

(4.) It is observable that in most windmills the velocity at 
the extremity of the sails is greater than that of the wind. In 
some cases, indeed, these velocities have been found in about 
the ratio of 5 to 2. Now, it is evident that the impulsion of a 
fluid against any surface whatever can only produce pressure, 
or mechanical effect, when the velocity of the surface exposed 
to the impulse is less than that of the fluid ; and that the pres- 
sure will be nothing when the velocity of the surface is equal 
to or greater than that of the fluid. Indeed, in the latter case, 
the pressure may operate against the motion of the sails, and 
be injurious. It is desirable, therefore, in order to derive from 
a windmill all the effect of which it is susceptible, so to adjust 
the number of the turns that the velocity of the extremity of 
the sails shall be less, or, at most, equal to that of the wind. 

It would be highly expedient to make comparative experi- 
ments on windmills, with a view to the determination of that 
velocity of the extremity of the sails which corresponds with 
the maximum of effect. 

6. If v denote the velocity of the wind in feet per second, t 
the time of one revolution of the sails, a the angle of inclination 
of the sails to the axis, and d the distance from the shaft or 
axle of rotation to the point which is not at all acted on by the 
wind, or beyond which the sail-cloth ought to be folded 
up ; then theoretical considerations supply the following theo- 
rem : viz. 

D = -1092 t v tan. A. 

Ex. Suppose v = 30 feet per second, t = 2 2 5 seconds, and 
a = 75° ; then 

d = -1092 X 30 x 2-25 x 3*73205 = 27'509 feet. 

This result agrees nearly with one of Coulomb's experi- 
ments, in which the velocity of the wind was 30 feet English 
per second, the sails made 17 turns in a minute, and they were 
obliged to fold off more than 6 feet from the extremity of 
each sail, of 34 feet long, to obtain a maximum of effect. 
The angle a at that distance from the tip of the sail was 75° 
or 76°. 



Section IV. — Steam and Steam-Engines. 

The whole power of the steam-engine depends on the em- 
ployment of elastic vapour produced from water at high tem- 
peratures. 

2h 2 



354 PNEUMATICS : STEAM AND STEAM-ENGINES. 

Steam, in fact, is highly rarefied water, the particles of which 
are expanded by the absorption of caloric, or the matter of heat. 
Water rises in vapour at all temperatures, though this is usually 
supposed to take place only at the boiling point ; when, how- 
ever, the evaporation occurs below 212° (Fahr.) it is confined to 
the surface of the fluid acted upon : but, at that heat, 212°, 
steam is formed at the bottom of the water, and ascends through 
it, carrying off the heat in a latent form, and, therefore, pre- 
venting the elevation of temperature of the water itself. At 
the common pressure of the atmosphere, one cubic inch of water 
produces about 1700 cubic inches (or nearly a cubic foot) of 
aqueous vapour ; but the boiling point varies considerably under 
different pressures, and these anomalies materially affect the 
the density of the vapour produced. Thus, in a vacuum water 
boils at about 70° ; under common atmospheric pressure at 212°; 
and when pressed by a column of mercury 5 inches in height, 
water does not boil until it is heated to 217° ; each inch of mer- 
cury producing by its pressure, a rise of about 1° in the ther- 
mometer. 

According to the elaborate experiments of Dr. Ure, of Glas- 
gow, the elastic force of this vapour at 212° is equivalent to the 
pressure of a column of mercury 30 inches high, or equal to 
about 15 lbs. avoirdupois on a square inch. 



temp. 212° 

226-3 . . 
238-5 . . 


. 30 inch. 

.40 

. 50-3 . . 


mercury 


15 lbs. per sq. inch. 

20 

25-15 


248-0 . . 
257*5 . . , 


. . 60 . . 
. 69-8 . . 




30 
34-9 


273-7 . . . 


. 91-2 . . 




45-6 


285-2 . . 


. 112-2 . . 




56-1 


312 


. 166* . . 




83-* 



And Mr. Woolf has ascertained that at these temperatures, 
omitting the last, a cubic foot of steam will expand to about 5, 
10, 20, 30, and 40 times its volume respectively ; its elastic 
force, when thus dilated, being in each case equal to the ordi- 
nary pressure of the atmosphere. 

One pound of Newcastle coals converts 7 pounds of boiling 
water into steam ; and the time required to convert a given 

* Some recent experiments made in France, by Messrs. Dulong and Arago, do 
not essentially differ in result from these of Dr. Ure. They find, at temp. 275* 18 
Fahr., an elasticity equal to 3 atmospheres, or 45 inches of mercury : at temps. 
308-84, 320-36, 331-70, 341-96,350-78, and 358-88, the elasticities equivalent to 
5, 6, 7, 8, 9, and 10 atmospheres, respectively. Temp. 439-34 an elasticity of 25 
atmospheres, which was the limit of their experiment ; but by computation they 
went to a temperature of 510-60, equivalent to an elasticity of 50 atmospheres. 



PNEUMATICS .* STEAM AND STEAM-ENGINES. 355 

quantity of boiling water into steam, is 6 times that required to 
raise it from the freezing to the boiling point. 

It is found, also, that if a bushel of coals per hour applied to 
a well-constructed boiler, produces steam of the expansive force 
of 15 lbs. per square inch, it will tend to expand itself with a 
velocity of 1340 feet per second ; then 2 bushels of coals, burnt 
under the same boiler, are capable of giving to the vapour an 
expansive force of 120 lbs. per square inch, and a velocity of 
expansion of 3800 feet per second. A bushel and half of coals 
would, with the same boiler, carry steam to the pressure of 
50 lbs. on a square inch ; which is as high as is regarded con- 
sistent with safety. 

From these data it will be evident that when steam is 
merely employed to displace the air in a close vessel, and after- 
wards produce a vacuum by condensation, no more heat is ne- 
cessary than what will raise the water employed to 212° : but 
if, on the contrary, steam capable of giving high pressures is 
required, a considerable increase of heat, as to 260°, 280°, will 
be necessary ; and, of course, an augmentation of fuel, though 
not one that is strictly proportional, will be required. This, 
however, is a consideration upon which we cannot here en- 
large. 

We proceed to speak of the actual construction of the ma- 
chine. 

The principles and manner of operation of the steam-en- 
gines of Savery, Newcomen and Cawley, and of Watt, may 
be understood from the following brief explanations and re- 
marks. 

1. Let there be a sucking pipe with a valve opening upwards 
at the top, communicating with a close vessel of water, not 
more than thirty-three feet above the level of the reservoir, 
and the steam of boiling water be thrown on the surface of the 
water in the vessel, it will force it to a height as much greater 
than thirty-three feet as the elastic force of the steam is greater 
than that of air ; and if the steam be condensed by the injection 
of cold water, and a vacuum thus formed, the vessel will be 
filled from the reservoir by the pressure of the atmosphere , 
and the steam being admitted as before, this water will also be 
forced up ; and so on successively. 

Such is the principle of the first steam-engine, said by the 
English to be invented by the Marquis of Worcester ; while 
the French ascribe it to Papin : though we believe the fact is 
that Brancas, an Italian, applied the force of steam ejected 
from a large oelopile as an impelling power for a stamping- 
engine so early as 1629. Brancas's was, in fact, an immense 



356 PNEUMATICS : STEAM-ENGINE j NEWCOMERS, &C. 

blow-pipe, turning a wheel. The hint so obscurely exhibited 
in the Marquis of Worcester's Century of Inventions, was car 
ried into effect by Captain Savery. 

2. If the steam be admitted into the bottom of a hollow cy- 
linder, to which a solid piston is adapted, the piston will be 
forced upwards by the difference between the elastic forces of 
steam and common air ; and the steam being then condensed, 
the piston will descend by the pressure of the atmosphere, and 
so on successively. This is the principle of the steam-engine 
first contrived by Messrs. Newcomen and Cawley, of Dart- 
mouth. This is sometimes called the atmospherical engine, and 
is commonly a forcing pump, having its rod fixed to one end 
of a lever, which is worked by the weight of the atmosphere 
upon a piston at the other end, a temporary vacuum being made 
below it by suddenly condensing the steam that had been ad- 
mitted into the cylinder in which this piston works, by a jet 
of cold water thrown into it. A partial vacuum being thus 
made, the weight of the atmosphere presses down the piston, 
and raises the other end of the straight lever, together with the 
water, from the well. Then immediately a hole is uncovered 
in the bottom of the cylinder, by which a fresh quantity 
of hot steam rushes in from a boiler of water below it, 
which proving a counterbalance for the atmosphere above the 
piston, the weight of the pump-rods, at the other end of the 
lever, carries that end down, and raises the piston of the 
steam-cylinder. The steam hole is then immediately shut, and 
a cock opened for injecting the cold water into the cylinder 
of steam, which condenses it to water again, and thus making 
a vacuum below the piston, the atmosphere again presses it 
down and raises the pump-rods, as before ; and so on con- 
tinually. 

3. When the cylinder is full of steam, if a valve be opened, 
by which the steam is allowed to escape into another vessel, 
where a jet of cold water is introduced, the condensation is 
much more complete, and the heat of the cylinder being pre- 
served, the steam possesses its full elasticity. 

This improvement was made by Mr. Watt, and completely 
changed the character of the steam-engine. In the old engines 
the power was diminished to half its real value, so that the 
moving force, instead of reaching 15 lbs. on each square inch 
of the area of the piston, was reduced to about 8 lbs. In this 
engine of Mr. Watt's the moving force is not less than 12 lbs. 
upon each square inch of the piston. 

4. A farther improvement has been made on this engine, 
by injecting the steam into the cylinder, alternately above 



PNEUMATICS : HIGH PRESSURE STEAM-ENGINE. 357 

and below the piston, so that the whole motion is produced by 
the elasticity of the steam, and has no dependance on the weight 
of the atmosphere. 

This improvement is also due to Mr. Watt, and could not 
have been made without the previous contrivance of condens- 
ing the steam in a separate vessel. It is particularly accom- 
modated to the production of a rotatory motion by means of 
a steam-engine. Three years before Mr. Watt introduced this 
improvement, viz. in 1778, Mr. Washborough, of Bristol, took 
out his patent for converting a reciprocating into a rotatory 
motion ; and in 1781, Mr. J. Steed effected the same thing, 
for the first time, by means of what is now called a crank. 
From that time Hornblower, Cartwright, Murray, Bramah, 
Trevithick, Maudslay, Woolf, and others, have, in rapid suc- 
cession, introduced a series of improvements which have 
rendered steam-engines as efficacious and perfect as can well 
be conceived. 

5. Another improvement due to Mr. Watt, is that of the 
Expansion Engine, invented about 1769. The principle of 
this invention, as Mr. Partington correctly remarks, consists 
in shutting off the farther entrance of steam from the boiler 
when the piston has been pressed down in the cylinder, for a 
certain proportion of its total descent, leaving the remainder 
to be accomplished by the expansive force of the steam al- 
ready produced.* To regulate the time of closing the valve, 
and as such the precise amount of steam admitted, Mr. Watt 
employed a plug-frame with moveable pins, which may be so 
placed that the steam-valve will shut when the piston has 
descended one-half, one-third, one-fourth, &c. By the appli- 
cation of this principle, the piston is made to descend uni- 
formly, the pressure on it continually diminishing as the steam 
becomes more and more rare, and the accelerating force is 
consequently diminished. 

6. The principle of the high-pressure steam-engine depends 
also on the power of steam to expand itself very considerably 
beyond its original bulk, by the addition of a given quantity 
of caloric, thus acquiring a considerable elastic force (equiva- 
lent to from 40 to 60 lbs. on each square inch) which, in this 
case, is employed to give motion to a piston. One of the 
greatest advantages attendant on employing the repellant 
force of steam, as in this form of the engine, consists in an 
evident saving of the water usually employed in condensation ; 
and this, in locomotive engines, for propelling carriages, is 
an object of considerable importance. The first description 

* See Brewster's Robison's Mechanical Philosophy, vol. iv. p. 126. 
47 



358 PNEUMATICS I HIGH PRESSURE STEAM-ENGINE. 

of an engine of this kind, which we have met with, is in Leo- 
pold's Theatre of Machines, published in Germany, in 1724. 
The apparatus consists of two cylinders placed at a moderate 
distance asunder ; each of them provided with a piston made 
to fit air-tight, and connected with a forcing-pump. When 
steam of considerable elasticity is admitted at the bottom of the 
first cylinder, it is forced upwards, carrying with it the lever 
of the pump ; at the same time that the steam or air is expel- 
led from the other. On this operation being repeated, or ra- 
ther reversed, the steam is allowed to enter the second cylin- 
der, which is also connected with the boiler, while the steam 
in the first cylinder is allowed to escape into the air. Thus, it 
may be remarked, that the process of condensation forms no 
part of the principle of the " high pressure" engine ; and that 
even the expansion of gunpowder might be employed to pro- 
duce a similar effect. 

7. We have already adverted to Mr. WoolPs discovery, that 
a quantity of steam having the force of 5, 6, 7, or more pounds 
on every square inch of the boiler, may be allowed to expand 
itself to an equal number of times its own volume when it 
would still have a pressure equal to that of the atmosphere, 
provided that the cylinder in which the expansion takes place 
have the same temperature as the steam possessed before it be- 
gan to increase. 

The most economical mode of employing this principle con- 
sists in the application of two cylinders and pistons of unequal 
size to a high pressure boiler ; the smaller of which should 
have a communication both at its top and bottom with the 
steam vessel. A communication being also formed between 
the top of the smaller cylinder, and the bottom of the larger 
cylinder ; and vice versa. When the engine is set to work, 
steam of a high temperature is admitted from the boiler to act 
by its elastic force on one side of the smaller piston, while the 
steam which had last moved it has a communication with the 
larger or condensing cylinder. If both pistons be placed at the 
tops of their respective cylinders, and steam of a pressure equal 
to 40 lbs. on the square inch be admitted, the smaller piston 
will be pressed down, while the steam below it, instead of being 
allowed to escape into the atmosphere, or pass into the con- 
densing vessel, as in the common engine, is made to enter the 
larger cylinder above its piston, which will make its downward 
stroke at the same time as that in the smaller cylinder ; and 
during this process, the steam which last filled the larger cy- 
linder will be passing into the condenser to form a vacuum 
during the downward stroke. 

To perform the upward stroke it is merely necessary to 



PNEUMATICS I HIGH PRESSURE STEAM-ENGINE. 359 

reverse the action of the respective cylinders ; and it will be 
effected by the pressure of the steam in the top of the small 
cylinder, acting beneath the piston in the great cylinder ; thus 
alternately admitting the steam to the different sides of the 
smaller piston, while the steam last admitted into the smaller 
cylinder passes regularly to the different sides of the larger pis- 
ton, the communication between the condenser and steam boiler 
being reversed at each stroke. 

Mr. Partington states that a double cylinder expansion en- 
gine of this kind was constructed for Wheal Vor mine in Corn- 
wall, in 1815. In this, the great cylinder is 53 inches in dia- 
meter, and has a nine feet stroke ; the small cylinder being in 
content about one-fifth of the great one. The engine works 6 
pumps, which at every stroke raise a load of water of 37,982 lbs. 
weight, 7i feet high. This produces a pressure of 14*1 lbs. per 
square inch on the surface of the great piston, while its average 
performance has been estimated at 46,000,000 lbs. raised one 
foot high with each bushel of coals. 

8. To render the preceding remarks and descriptions more 
intelligible, we will now give fuller descriptions of a few en- 
gines : — Plate II. fig. 1, exhibits a vertical section of the dif- 
ferent parts of a high pressure steam-engine, constructed upon 
a principle in which simplicity and power are blended as far 
as possible ; and in which the parts are arranged in such a 
manner as seemed best calculated to facilitate the comprehen- 
sion of these machines to such as have not already had an op- 
portunity of examining them carefully. The construction is 
due to Oliver Evans ; and every thing not essentially neces- 
sary in a popular account of the operation, is omitted in the 
drawing. 

a, section of the boiler, perpendicular to its axis. It con- 
sists of two cylinders, one within the other, (the cylindric 
form being most capable of resisting a great expansive force.) 
The fire is made to burn in the interior cylinder which serves 
for the furnace, and the water is contained between the two 
cylinders. The smoke and heated air, after having traversed 
the interior cylinder in the whole direction of its length, 
passes out at the other extremity, and is thence conducted 
under the elementary or subsidiary boiler, b, where it heats 
the water which is intended to supply in the great boiler the 
place of that which has been carried off in the state of vapour, 
and thus given motion to the machine, c, the feeding pump, 
which at every stroke of the piston, and corresponding motion 
of the great beam, raises a small quantity of cold water, and 
forces it into the supplementary boiler. The object of this 
part of the apparatus may be explained in few words. If the 



360 

waste of water in the great boiler, by its conversion into steam, 
were supplied at once from the cold water pump, the tem- 
perature of the liquid in the great boiler would be diminished, 
and the production of steam proportionally checked. But 
when water passes from the pump to the great boiler, through 
the intervention of the subsidiary boiler in which it becomes 
heated, it enters the great boiler without diminishing the tem- 
perature of the water which it contains, and, of course, without 
checking the operation of the machine. The skill and judg- 
ment of the engineer are evinced in proportioning the magni- 
tudes of the two boilers, and of the feeding pump, so that the 
supply may be neither defective nor superabundant, but just 
sufficient. 

The steam rises through the tube, and if the injection 
sucker, d, is opened to permit the entrance of steam into the 
machine, the valves e and f being opened, the steam will push 
the piston g to the inferior extremity of the cylinder ; as is 
shown in the diagram. The steam which occupied the cylin- 
der, being driven before the piston, escapes through the valve 
f. As soon as the piston g arrives at the bottom of the cylin- 
der, the valves e and f close, and the other valves h and i 
open ; the steam enters at h, and causes the piston to rise, 
while the weaker steam, which is contained in the upper part 
of the cylinder, escapes through the valve i, presenting scarcely 
any reaction to the ascent of the piston. These four pistons 
are moved by two wheels k and l, carrying cams or lifters on 
their surfaces, which press against four levers, to which stems 
or handles are attached, and which thus cause them to open 
and shut precisely at the adequate intervals. [These levers 
are not represented in the figure, but are omitted to prevent 
confusion.] 

The motion of the piston g, of course, communicates a re- 
ciprocating motion to the beam m n ; and the crank and con- 
necting bar m o give a rotatory motion, the regularity of which 
is maintained by means of the fly-wheel q r. After similar 
operations, the valves e, f, h, i, alternately opening and shutting, 
the motion of the machine is continued. The teethed wheels 
s, t, of equal diameters, give motion to the wheels and lifters 
l, k ; and the beam m n communicates the requisite motion to 
the forcing piston of the feeding pump c. The motion of the 
whole being thus continued ; the wheel v, if it have 66 teeth, 
playing into the wheel u of 23 teeth, would give, say to the 
mill-stone w, 100 revolutions for every 35 strokes of the pis- 
ton g : viz. 100 revolutions in a minute if the piston make 35 
strokes in that interval of time. 

The toothed wheel v may be applied to any other work. 



PNEUMATICS I OLIVER EVANs's STEAM-ENGINE. 361 

Instead of the wheel, indeed, there may be employed a crank 
to move a pump, or a saw ; and the machine may, by modify- 
ing the power of the steam, give from 10 to 100 strokes of the 
piston per minute, according to the purpose for which it is 
constructed. If the steam cylinder be 8 inches in diameter it 
will work a mill-stone of 5 feet diameter ; or perform any 
work which requires equal mechanical energy. 

The steam, on quitting the machine, escapes by the tube 
x x, and becomes dissipated in the air above the top of the 
building : or, if the constructor please, it may pass into the 
condenser, when a condenser is used ; or it may be made to 
pass along the subsidiary boiler, to heat the water which it 
contains. 

y, is the safety-valve, kept in its place by a lever, graduated, 
like the steelyard, to weigh and thus balance the effort of the 
steam. This valve rises and permits the excess of steam to 
escape when its elastic force becomes more considerable than 
is requisite for the work of the machine. The weight is, in 
general, so adjusted upon the lever, that the valve shall open 
when the pressure of the steam upon a square inch of the in- 
terior of the boiler becomes about half that for which its thick- 
ness and strength of material are fitted. 

To prevent a recurrence of those accidents which have 
sometimes occurred aboard steam-boats, and which first drew 
the attention of the legislature to this important part of the 
engine, it appears advisable to enclose the safety-valve in an 
iron box, and thus place it beyond the control of the engine- 
man. 

The whole of the machine now described is very simple in 
its construction, and easy to execute by ordinary mechanics or 
mill-wrights. The orifices for the passage of the steam are 
simple metallic plates, each pierced with a hole, on which the 
valve intended to close it readily falls : they are very easily 
cast at any common foundry. 

9. Manufactories of steam-engines are now numerous, es- 
pecially in those districts where machinery is much called 
for. The engines of Maudslay are much celebrated, especially 
in situations where it is of consequence to erect an engine in 
small space, as in steam-boats, &c. Next to Maudslay's, the 
engines of Messrs. Fenton, Murray, and Wood, of Leeds, have, 
perhaps, attained the highest celebrity. As the construction 
of their engines serves well for the elucidation of the manner 
of operation, we here present a description of one of them, 
from a very useful work, Smith's Panorama of Science and 
Art. 

In Plate III, a represents the boiler, nearly three quarters 

£2 I 



362 PNEUMATICS ! FENTON AND CO. S STEAM-ENGINE. 

full of water : the bottom is considerably, and the sides a little 
concave, that it may receive more fully the force of the flame 
circulating round it. Boilers are usually of an oblong form, 
and are furnished with a part that takes off, in order that a 
person may get in to clean them when needful ; they have also 
a valve, called the safety-valve, opening upwards, which is 
loaded so that the steam escapes when it is stronger than the 
engine requires, and, if retained, would hazard the bursting of 
the boiler. It is not uncommon to have two boilers, one of 
which is a reserve, that the engine may not be stopped, when 
the other requires repair. 

b, is an apparatus for regulating the fire, and giving action 
to a bell, which regulates the quantity of coals and time of 
firing. 

c, the steam-pipe from the boiler a to the valve i. 

d, the steam-cylinder, generally called only " the cylinder ;" 
it is connected at the top and bottom with the valve i. 

e, the piston, which, by its connecting rod e, gives motion 
to the beam f, the other end of which, by another connecting 
rod, gives motion to the heavy fly-wheel g, by means of a 
crank. Thus, after the engine has begun to work, its power 
is accumulated in the fly-wheel, and may be disposed of at the 
pleasure of the mechanist. 

h, an eccentric circle* on the axle of the fly-wheel g : it gives 
motion by its levers, in a manner easily understood by inspec- 
tion, to the valve i. 

i, a coffer-slide valve, which requires no packing to make it 
steam-tight, as there is always a vacuum under it : it answers 
the purpose of the four valves used in double-power engines, 
and from the simplicity of its construction, when well made at 
first, is not liable to get out of order. 

k, the steam-admission valve and lever, connected with a 
governor not shown in the figure, which regulates the speed 
of the engine. See p. 262. 

l, the cylinder of the discharging pump, for extracting the 
water and uncondensed vapour from the condenser m. 

n, a small cistern, filled with water. Into this cistern enters 
a pipe from the condenser m, the top of which pipe is covered 
by a valve, which is called the blow-valve, or sometimes the 
snifting-valve. Through this valve, the air contained in the cy- 
linder d, and passages from it, is discharged, previously to the 
engine being set in motion. 



* Mechanics distinguish by the appellation of " eccentric circle," a wheel, the 
axle of which is purposely made to pass, not through its real centre, but on one 
side of it. 



o, the eduction-pipe, which conducts the steam from the 
valve i to the condenser m. 

p, the pump which supplies with water the cistern, ss, in 
which the condenser and discharging-pump stand. 

qq, iron columns, of which the engine has four, although only- 
two are shown ; they stand upon one entire plate, seen edge- 
way, on which the principal parts of the engine are fixed ; by 
this means the beam and its accompaniments are supported 
without being connected with any part of the building, except 
the recess below the floor on which they stand. 

r r, the recess below the floor, for containing the cistern of 
the discharging-pump, condenser, &c. This arrangement ena- 
bles those engines to be fixed up and tried at the manufactory 
before they are sent off, which renders the refixing easy and 
certain. Engines are made according to this plan, from the 
power of one to twelve horses. 

Before the engine is set to work, the cylinder d, the con- 
denser m, and the passages between them, are filled with com- 
mon air, which it is necessary to extract. To effect this, by 
opening the valves, a communication is made between the 
steam-pipe c, the space below the piston in the cylinder d, the 
eduction-pipe o, and the condenser m. The steam will not at 
first enter the cylinder d, or will only enter it a little way, be- 
cause it is resisted by the air ; but the air in the eduction-pipe o, 
and the condenser m, it forcibly drives before it, and this part 
of the air makes its exit through the valve and water in the cis- 
tern n. The steam-admission valve is now closed, and the 
steam already admitted is converted into water, partly by the 
coldness of the condenser m, but principally by a jet of cold 
water which enters it through a cock opening into it from the 
well ss, in which the condenser is immersed. When this steam 
is condensed, all the space it occupied would be a vacuum, did 
not the air in the cylinder d expand, and fill all the space that 
the original quantity of it filled ; but by the repetition of the 
means for extracting a part of the air, the remainder is blown 
out, and the cylinder becomes filled with steam alone. Sup- 
pose then the cylinder beneath the piston to be filled with 
steam, and the further admission of steam to that part of it be 
cut off, while the communication between it and the condenser 
remains open, it is obvious that there will soon be a vacuum in 
the cylinder, because as fast as the steam reaches the condenser, 
it is converted into water by the coldness of that vessel and the 
jet playing within it. At this moment, therefore, the steam is 
admitted above the piston, which it immediately presses down. 
As soon as the piston reaches to the bottom of the cylinder, the 
steam is admitted to the under side of it, and as the communi- 



364 PNEUMATICS I FENTON AND CO. S STEAM-ENGINE. 

cation from the upper side of the piston to the condenser is 
opened, while the further admission of steam to that side 
during the upper stroke is prevented, the steam which had 
pressed the piston down passes into the condenser, and is con- 
verted into water. 

The motion of the piston e, by this alternate admission and 
extraction of the steam on each side of it, is thus necessarily- 
continued, and the distance of its upward and downward range 
is called the length of its stroke. It communicates its recipro- 
cating motion, by the connecting rod e, to the great beam r, 
and thence, by another connecting rod and a crank, to the fly- 
wheel G. 

To explain the rapid accumulation of power with an increase 
of the size of the engine, it must be observed, that the force of 
the steam generally used, is somewhat greater than the pressure 
of the atmosphere ; but supposing it to be no greater, as the 
atmospheric pressure is fifteen pounds on each square inch, a 
piston sixteen inches in diameter, containing 201 square inches 
of surface, will alternately be raised and depressed by a force 
equivalent to a weight of 3015 pounds. Here no allowance is 
made for friction, but after the requisite deduction on this ac- 
count, which may be reckoned at one-third, the disposable part 
of the engine, derived from each stroke, will still be very 
great. 

The condenser m, and the discharging-pump l, communicate 
by means of a horizontal pipe containing a valve y opening 
towards the pump ; the piston, /, of this pump, also contains 
two valves, and the cistern t, at the top of the pump-cylinder, 
contains two other valves,, which, like those of the piston /, 
open upwards. When the piston e of the cylinder is de- 
pressed, the piston /, of the discharging-pump, it will be obvious 
to inspection, is depressed likewise, and its valves open, while 
the valve y closes ; hence the water from the condensed steam, 
as well as the injection-water, and any permanently elastic va- 
pour or gas, which may be present, having passed through the 
valve y, passes through the piston / ; and when that piston is 
drawn up, its valves close and prevent their return, as in ordi- 
nary pump-work. The water and gas that have thus got above 
the piston, as the latter rises, open the valves at the bottom of 
the cistern t, in which the water remains till it is full, but the gas 
passes into the atmosphere. As the water in the cistern t is in a 
very hot state, it is sometimes, for the purpose of economizing 
fuel, pumped up and returned to the boiler, the pump-rod being 
attached to the great beam. The utility of the discharging-pump l 
will now be appreciated, and it must be perceived how much more 



PNEUMATICS : FENTON AND CO.'s STEAM-ENGINE. 365 

materially it contributes to the perfection of the vacuum in the 
cylinder d, than if the water from the condenser merely ran 
off by a pipe. 

The steam constantly rushing into the condenser m, has a 
perpetual tendency to heat that vessel, as well as the water of 
the cistern s s, in which it stands ; the whole of the steam, 
if this were unchecked, would not be condensed, or the con- 
densation would not be sufficiently rapid, because the injec- 
tion-water itself flows out of this cistern. A part of the water 
is therefore allowed to flow from this cistern by a waste pipe, 
and an equal quantity of cold is constantly supplied by the 
pump p. 

In Newcomen's, or the atmospherical engine, the cylinder 
was open at the top, and therefore, during the descent of the 
piston, the air exerted a great power in cooling it ; but in the 
modern engines, where steam is the active power both in 
raising and depressing the piston, the top of the cylinder is 
closed with an iron lid, and not an atom of steam can escape, 
except at the proper time, into the condenser. In order that 
the connecting rod e may work freely, and yet possess this de- 
sirable property of being steam-tight, it passes through what 
is called a stuffing or packing box. This stuffing consists of 
some material which the steam will rather adapt to its office 
than injure ; leather, which is used for the stuffing or collars 
of machines never to be subjected to heat will not answer 
here ; hempen yarn is the material usually employed. The 
rod of the piston /, passes through a stuffing box of the same 
kind as that of the piston e ; and the pistons themselves are 
surrounded with stuffing. 

The cylinder d is surrounded by a case, to keep it from 
being cooled by contact with the external air. The extre- 
mity, or any given point removed from the centre of the great 
beam, can describe only the arc of a circle ; but it is neces- 
sary that the piston rod e should rise and fall vertically. New- 
comen effected this object, by making the end of the beam 
into the arc of a circle, the radius of which was equal to the 
distance from the centre of the beam : a chain went over this 
arc, and was fastened on the higher end of it ; this simple 
contrivance effectually answered his purpose, because in his 
engine the effective stroke was only downwards ; but here, in 
a double-power engine, where the stroke is both upwards and 
downwards, a chain would yield in rising, and be altogether 
unsuitable. An apparatus is therefore used, called the pa- 
rallel joint, which is easily understood by inspection. By 
this means the rod e not only rises and falls perpendicularly, 
but is perfectly rigid, and communicates all its motion to the 
48 2 i 2 



366 PNEUMATICS : FENTON AND CO.'s STEAM-ENGINE. 

great beam in each direction of its motion. The connecting 
rod g does not require the same contrivance, because it does 
not rise and fall perpendicularly ; its lower end, with the outer 
end of the crank, describing a circle : it has therefore only a 
simple joint, admitting of this deviation. 

In order to communicate a rotatory motion to the fly-wheel, 
instead of the crank may be used a contrivance giving twice 
the rapidity to the fly. For this purpose, on the outside of 
the axis of the fly, where the crank is shown in the plate, a 
small toothed wheel is fixed, and can only be moved with the 
fly : at the extremity of the rod g, and on that side of it which 
is next the fly-wheel, another toothed wheel is fixed, in such a 
manner that it cannot turn round on its axis, but must rise and 
fall with the rod to which it is attached. These two wheels 
work in each other, and that attached to the connecting rod 
cannot leave its fellow, because their centres are connected by 
a strap or bar of iron. When, therefore, the connecting rod 
rises, the wheel upon it moves round the circumference of the 
wheel upon the axis of the fly. By this means the fly makes 
an entire revolution for every stroke of the piston, and some 
mechanics are apt to think that they are great gainers by such 
an arrangement : the contrivance is certainly elegant ; but 
with respect to utility, the fact is, that a crank is preferable ; 
for it is more simple, cheaper, and less likely to be out of order, 
while, if the fly be large enough to receive, with less velocity, 
all the momentum that can be communicated to it, the effect 
will certainly not be inferior. 

10. Locomotive steam-engines, or those which will propel 
both themselves when placed upon wheels, and any suitable 
carriages attached to them, were invented by Mr. Trevithick, 
in Cornwall, about the year 1804. They are now in constant 
use in the northern coal districts ; their peculiar construction 
will be very evident from the following description, by Mr. 
Tredgold, of the engines employed on the Hetton rail-way. 

The wheels of the coal-wagons drawn by the engines are 
2 feet 11 inches in diameter, with 10 spokes, and weigh 2% 
cwt. Their axles are 3 inches in diameter, and revolve in 
fixed bushes. 

The weight of each engine is about 8 tons. It consists of 
a boiler 4 feet in diameter, with an internal fire-place. The 
smoke ascends from the fire by a chimney about 12 feet high ; 
the lower 3 feet of the chimney is formed of sheet iron of 6 
lbs. to the square foot, and the rest of iron 2\ lbs. to the 
foot. There are two cylinders, which work alternately. The 
diameter of the pistons is 9 inches, and the length of the 
stroke 2 feet ; the pistons make about 45 double strokes per 



PNEUMATICS : LOCOMOTIVE STEAM-ENGINE. 367 

minute. The steam is admitted to the cylinders by slide 
valves worked by eccentric wheels on the axis of the engine 
carriage. The pressure of the steam in the boiler is from 40 
to 50 lbs. on the square inch. 

The wheels of the engine carriage are 3 feet 2 inches dia- 
meter, with 12 spokes in each, and each weighs 3| cwt. The 
axles are 3§ inches diameter, and are connected by an endless 
chain working into a wheel on each axle, so that both the 
axles of the carriage may be turned at the same rate. The 
boiler is supported on the carriage by four floating pistons, 
which answer the purpose of springs in equalizing the pres- 
sure on the wheels, and softening the jerks of the carriage. A 
floating piston is packed as the steam piston of a steam-engine, 
and has a short piston rod of li inches diameter, which rests 
upon the brass bush, in which the axle of the wheel turns. 
The water in the boiler presses on the upper side of the piston ; 
and whatever elevation or depression the wheel follows, the 
pressure upon it is nearly the same. This ingenious substi- 
tute for a spring, as well as the other peculiarities of this engine, 
were invented by Messrs. Losh and Stephenson, of New- 
castle-upon-Tyne, and made the subject of a patent in 1816. 

Fig. 2. plate II. is a sketch of the steam-carriage, &c. a is 
the boiler, and b b the steam cylinders ; the fireplace is within 
the boiler, and f is the entrance to it ; c is the chimney : d d 
the floating pistons which support the carriage on the axles, 
and answer as springs in making it press equally on the rails. 
As the moving force is not equal at the same time on the 
wheels of both axles, it is necessary to connect the axles by a 
pitch chain g, working into toothed wheels on the axles. The 
water for supplying the boiler, and the coals at b for the fire, 
are carried by a small carriage called the tender ; i is the water 
barrel, and a is a hose pipe which conveys the water to the 
force-pump h, which is worked by the engine ; w w are coal 
wagons, each of which carries 53 cwt. of coals. From 13 to 
17 of these wagons are drawn in a train by one steam-carriage ; 
they are connected by the short chains c c. The connecting rods 
which communicate the power from the pistons to the wheels 
of the steam-carriage are attached to the wheels, so that one pis- 
ton is at half the length of its stroke, when the other is at the 
commencement of its stroke. 

Fig. 3, is a vertical transverse section of the steam-carriage ; 
the comparison of which with fig. 2, will render the whole more 
intelligible. 

11. We shall terminate our description of steam-engines, 
by exhibiting one invented by Mr. John Nuncarrow, an 
American, and which is intended to give motion to water- 



368 PNEUMATICS : NUNCARROW's STEAM-ENGINE. 

wheels, in places where there is no fall, and but a very small 
stream or spring. 

a, fig. 4, plate II, the receiver, which may be made either of 
wood or iron. 

b, b, b, b, b, wooden or cast iron pipes for conveying the 
water to the receiver, and thence to the penstock. 

c, the penstock or cistern. 

d, the water-wheel. 

e, the boiler, which may be either iron or copper. 

f, the hot well for supplying the boiler with water. 

g, g, two cisterns, under the level of the water, in which the 
small bores b, b, and the condenser are contained. 

h, h, h, the surface of the water with which the steam-engine 
and the water-wheel are supplied. 

a, a, the steam-pipe, through which the steam is conveyed 
from the boiler to the receiver. 

b, the feeding-pipe, for supplying the boiler with hot water. 

c, c, c, c, c, the condensing apparatus. 

d, d, the pipe which conveys the hot water from the condenser 
to the hot well. 

e, e, e, valves for admitting and excluding the water. 

f, f, the injection pipe, and g, the injection cock. 
h, the condenser. 

It must be remarked that the receiver, penstock, and all the 
pipes, must be previously filled before any water can be deli- 
vered on the wheel ; and when the steam in the boiler has ac- 
quired a sufficient strength, the valve at i is opened, and the 
steam immediately rushes from the boiler at e into the receiver 
a ; the water descends through the tubes a and b, and ascends 
through the valve k, and the other pipe or tube b, into the pen- 
stock c. This part of the operation being performed, and the 
valve i shut, that at e is suddenly opened, through which the 
steam rushes down the condensing pipe c, and in its pas- 
sage meets with a jet of cold water from the injection 
cock g, by which it is condensed. A vacuum being made 
by this means in the receiver, the water is driven up to fill it a 
second time through the valves e, e, by the pressure of the ex- 
ternal air, when the steam-valve at i is again opened, and the 
operation repeated for any length of time the machine is re- 
quired to work. 

There are many advantages which a steam-engine on this 
construction possesses beyond any thing of the kind hitherto 
invented ; a few of which the inventor thus enumerates. 

(1.) It is subject to little or no friction. 

(2.) It may be erected at a small expense when compared 
with any other sort of steam-engine. 



PNEUMATICS : POWER OP STEAM-ENGINES. 369 

(3.) It has every advantage which may be attributed to Boul- 
ton and Watt's engines, by condensing out of the receiver, 
either in the penstock or at the level of the water. 

(4.) Another very great advantage is, that the water in the 
upper part of the pipe adjoining the receiver acquires a heat, 
by its being in frequent contact with the steam, very nearly 
equal to that of boiling water ; hence the receiver is always 
kept uniformly hot, as in the case of Boulton and WatVs en^ 
gines. 

(5.) A very small stream of water is sufficient to supply this 
engine (even where there is no fall), for all the water raised by 
it is returned into the reservoir h, h, h. 

From the foregoing reasons it would seem that no kind of 
steam-engine is better adapted to give rotatory motion to ma- 
chinery of every kind than this. Its form is simple, and the 
materials of which it is composed are cheap ; the power is more 
than equal to any other machine of the kind, because there is 
no deduction to be made for friction, except on account of turn- 
ing the cocks, which is but trifling. 

Its great utility is therefore evident in supplying water for 
every kind of work performed by a water-wheel, such as grist- 
mills, saw-mills, blast-furnaces, forges, &c* 

12. The theory of steam-power in reference to the mechani- 
cal energy of engines, is, as yet, in a very imperfect state. 
The best formulae which we have hitherto seen, are exhibited 
by the late Mr. Tredgold, in his judicious and valuable work 
on Rail-Roads. As they are found to furnish results which 
agree very nearly with those of experiment, we insert them 
here, with the author's permission. 

If / be the measure of the force of steam in inches of the 
mercurial column, and t the corresponding temperature mea 
sured on Fahrenheit's thermometer ; f the resistance from the 
friction of the steam piston, and the condensed vapour in the 
cylinder, or the atmospheric pressure in high-pressure engines, 
and n the bulk or capacity of the steam cylinder, when the 
bulk of the steam admitted at the pressure f is unity. Then 
the power of the steam generated from a cubic foot of water is 

4873 (459 + t) X (l — ^£-)+ hyp. log. n). 
When the steam does not act by expansion n = 1. 

* For descriptive information on the subject of steam-engines, the reader may 
consult the historical treatises of Partington and Stuart ; and, for still more com- 
prehensive and scientific accounts, the treatises of Tredgold and J. Farey. For 
some of the recent improvements by Hall and others, see the Mechanics' Maga- 
zine, vol. 18. 



370 pneumatics: steam-engine power. 

When the expanding force of the steam is employed, the 
above equation has a maximum, which will obtain when hyp. 

nf ... 
log. n — is a minimum, which is evidently the case when 

n = — . In that case, inserting, ^-, for n, we have 
j j 

f 
4873 (459 -f /). (hyp. log. j-) = the maximum power of a 

cubic foot of water converted into steam. 

f 
Whenf = f, then hyp. log. j- t = 0, and the power is no- 

«/ 
thing. 

f f 

And, when 1 — J — is greater than hyp. log. j- t it is disad- 
vantageous to work by expansion. 

1 3. To calculate the quantity of fuel, let c be the quantity 
which converts a cubic foot of water into steam that will bear 
the pressure of the atmosphere ; let s be the specific heat of 
the steam, a the specific heat of the air and smoke which 
escape up the chimney, and w the weight of fuel that will 
heat one cubic foot of water one degree : then 

c + [ (t — 212°) x (a + s) to] = 
the least quantity of fuel that will produce steam of the force 
/ and temperature /. 

Mr. Tredgold,by assuming c = 8*4 lbs. of Newcastle coals, 
w = '0075 lbs. s = '847, and a = *753, reduces the preceding 
to 8*4 + -012 (/ — 212°) = the lbs. of coal to produce steam 
of the temperature t. 

14. For a high-pressure engine, taking 30 inches for the 
measure of atmospheric pressure, i of the pressure of the 
steam for the friction of the steam piston, and -^ f for the 
plus pressure in the boiler, the whole loss becomes J f. But, 
one side of the piston of a high-pressure engine is acted 
upon by the same pressure as that of the external atmos- 
phere : hence f = ^ f -f 30 = the resistance to the moving 
force f. 

Consequently, when a high-pressure engine is worked ex- 
pansively, we have 

4873 (459 + t) X (hyp. log.j-^-^)= 

the mechanical power of a cubic foot of water converted into 
steam. 

Hence, there is no advantage in making a high-pressure 



PNEUMATICS .* STEAM-ENGINE POWER. 371 

steam-engine work expansively, when the force of the steam is 
less than 60 inches of the mercurial column ; because the above 

hyp. log. is then less than 1 — f 

When an engine does not employ the expansive power of 
steam, we have 

4873 (459 + 4) X (l — *^"JT 3 ° ) = the mechanical power 

of a cubic foot of water converted into steam. 

15. Mr. Tredgold illustrates these formulae by the follow- 
ing : 

Example. Let the force of the steam be 120 inches of mer- 
cury ; the corresponding temperature is 292*8°. Then 

4873 (459 + 292*8) x(l — ^ X 120) + 3 ° = 

1,830,000 lbs. raised one foot high. 

The quantity of coal is 8-4 -f '012 (292*8 — 212) = 9*37 lbs. 
of coal. 

Now if the horse power be 16,000,000 lbs. raised one foot in 
a day of 8 hours ; then 

1,830,000, : 9-37 lbs. :: 16,000,000 : 82 lbs. 

Therefore, working with steam of 44§ lbs. on the square 
inch on the piston, above the pressure of the atmosphere, 
82 lbs. of Newcastle coal ought to do the day's work of a 
horse. 

But if the engine works expansively with the same force of 
steam, then 

4873 (459 + 292*8) X hyp. log. 2) = 2,540,000 lbs. 
raised one foot high by 9*37 lbs. of coal ; and consequently 
59 lbs. of coal ought to do the day's work of a horse. 

16. With regard to the maximum of useful effect in steam- 
engines, it will be found, according to Mr. Tredgold, by taking 
v = 120 y/ /, for the working velocity of an engine in feet per 
minute, / being the length of the stroke in feet. 

If an engine has a 2 feet stroke, then v = 170 feet per mi- 
nute, and the number of strokes per minute 42^. 

By increasing the stroke to 3*4 feet, we get a velocity of 220 
feet per minute, with 32 strokes per minute. 

If any variation be made from the maximum power, the de- 
crease of effect is the same as in horse power *. but, as Mr. T. 
remarks, we have this advantage in an engine — it can be made 
for any velocity, by attending to the relative proportions of its 
parts ; those of a horse we cannot alter. 

17. A horse, when he treads a mill-path at the rate of 2§ 



372 PNEUMATICS : STEAM-ENGINE POWER. 

miles an hour, will, on an average, raise about 150 lbs. by a cord 
hanging over a pulley ; which is equivalent to 33,000 lbs. one 
foot high in a minute. Boulton and Watt estimate this at 
32,000. Tredgold, still lower, at 27,500. Taking the first 
measure, however, as a basis of comparison ; putting d for the 
diameter of the piston in inches, p for the pressure of the steam 
upon each square inch (diminished usually by about ~ for fric- 
tion and inertia), / for the length of the stroke of the piston in 
feet, n for the number of strokes in a minute : then the power 
of the engine in " Horse-powers" (h p), is 

(h p) = -0000238 d 2 n p I, if it be a single stroke £ 
h p = -0000476 d»np /, if it be a double stroke $ en S ine ' 

Example. Suppose d = 20 inches, 1=3 feet, n = 36, p = 
50, and the engine one of double stroke. Then 

•0000476 x 20 3 X 36 x 50 X 3 == 102-816, or nearly 103" 
horse-powers, the measure of the energy of the engine. 

Mr. Boulton states that 1 bushel of Newcastle coals, contain- 
ing 84 pounds, will raise 30 million pounds 1 foot high ; that it 
will grind and dress 11 bushels of wheat ; that it will slit and 
draw into nails 5 cwt. of iron ; that it will drive 1000 cotton 
spindles, with all the preparation machinery, with the proper 
velocity ; and that these effects are equivalent to the work of 
10 horses. 

18. The rule usually given to adjust the weight of the fly- 
wheel is this : 

Multiply the number of Horse-powers in the machine by 
2000 ; divide the product by the square of the velocity in feet, 
per second, of the fly's circumference ; the quotient will give its 
weight in hundred-weights. 

Or, 2000 (h ?) ^-^y = w\ of fly. 

Thus, suppose the fly-wheel of a 20 horse-power engine to be 
18 feet diameter, and to revolve 22 times in a minute ; what 
should be its weight ? 

„ 18 X 3-1416 X 22 . . 

Here — = 20i feet nearly, velocity of cir- 
cumference per second. 

Whence — - — — — = 90*4, weight of the fly-wheel re- 
(2O2) 

quired. 

:* See, farther, p. 262, preceding. 



* 



STEAM-ENGINES, CANALS, RAIL-ROADS, &C. 



373 



CHAPTER XIV. 



USEFUL TABLES AND REMARKS ON STEAM-ENGINES, RAIL-ROADS, 
CANALS, AND TURNPIKE-ROADS. 

Table I. — Quantity of Coals equivalent to the horsepower 
of 33,000 lbs. raised one foot per minute in high pressure 
steam-engines ,when the greatest possible effect isobtained* 



O 
0) 

h 

I- 

Eh 


II 

rt id 


MS 

jas a. 

"OB 

c.S| 

o sS 
fa =*5 

^ on ** 


Quantity of coal equivalent to 
a horse power. 


Pounds raised one foot high 

equivalent to the immediate 

power of the steam produced 

by 84 lbs. of coal. 


When working 
at full pressure. 


When acting 
expansively. 


When work- 
ing at full 
pressure. 


When work- 
ing expan- 
sively. 


o 
234-5 
251 
275 
292-8 
307-7 
320-2 
343-6 


inches. 

45 

60 

90 
120 
150 
180 
240 


lbs. 
7-4 

14-8 
29-7 
44-5 
59-3 
74-2 
M)4 


lbs. 
480 
163 
98 
82 
74 
70 
65 


lbs. 

143 

77 
59 
51 
48 
41* 


lbs. 
2,780,000 
8,200,000 
13,700,000 
16,600,000 
18,000,000 
19,200,000 
20,500,000 


lbs. 

9,300,000 
17,700,000 
22,700,000 
26,200,000 
28,700,000 
32,200,000 



Table II. — Quantity of Coals equivalent to the horse power 
of 33,000 lbs. raised one foot per minute in condensing 
steam-engines, when the greatest possible effect is obtained. 



«M 


>» 


O) 




Pounds raised one foot high 


o 


c s 


<D O 0J 


Quantity of coal equivalent to 


equivalent to fche immediate 


g 


Z s 


tS» 


a horse power. 


power of the steam produced 


*1 

6 


6»a 


o s 2 
fa ^5 




by 84 lbs. of coal. 


When working 
at full pressure. 


When acting 
expansively. 


When work- 
ing at full 
pressure. 


When work- 
ing expan- 
sively. 


o 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


220 


35 


2-5 


63£ 


40£ 


21,000,000 


33,100,000 


234-5 


45 


7-4 


62 


38£ 


21,400,000 


35,200,000 


251 


60 


14-8 


60 


35£ 


22,400,000 


37,500,000 


275 


90 


29-7 




33£ 




40,000,000 


292-8 


120 


44-5 




32§ 




41,000,000 


307-7 


150 


59-3 




32 




42,400,000 


320-2 


180 


74-2 




31$ 




42,700,000 


343-6 


240 


104-0 4 


31 




43,500,000/ 



* The curious tables here given, marked I. II. III., were extracted, with the au- 
thor's permission, from Mr. Tredgold's work on Rail-roads. 
49 2 K 



374 



STEAM-ENGINES, CANALS, RAIL-ROADS 



Remarks on Tables I, and II. — The columns showing the 
pounds an engine ought to raise one foot high, by the heat of 
one bushel of coals, are added chiefly for the purpose of com- 
parison with actual practice. Now, it is stated, that after the 
most impartial examination for several years in succession, it is 
found that Woolf 's engine at Wheal Abraham Mine, raised 44 
millions of pounds of water, one foot high, with a bushel of 
coals. And, " the burning of one bushel of good Newcastle or 
Swansea coals, in Mr. Watt's reciprocating engines, working 
more or less expansively, was found, by the accounts kept at 
the Cornish mines, to raise from 24 to 32 millions of pounds of 
water one foot high ; the greater or less effect depending upon 
the state of the engine, its size, and rate of working, and the 
quality of the coal." 

We shall further add the results of half a year's reports taken, 
without selection, from Lean's Monthly Reports on the work 
performed by the steam-engines in Cornwall, with each bushel 
of coals. The numbers show the pounds of water raised one 
foot high with each bushel, from January to June, 1818. 



22 to 25 Common En- 
gines average 
Wheal Vor (Woolf's 

Engine) 
Wheal Abraham (ditto) 
Ditto (ditto) 

Wheal Unity (ditto) 
Dalcouth Engine "' 
Wheal Abraham Engine 
United Mines Engine 
Treskirby Egnine 
Wheal Chance Engine 



January. 



one foot. 
22,188,000 

30,834,000 

41,847,000 
27,942,000 
31,900 ; 000 
42,622,000 
32,239,000 
36,396,000 
38,733,000 
28,496,000 



February. 



lbs. raised 
one foot. 

22,424,000 

26,158,000 

35,364,000 
'28,000,000 
32,306,000 
41,354,000 
36,180,000 
31,830,000 
39,375,000 
32,319,000 



March. 



bs. raised 
one foot. 

21,898,000 

29,511,000 

30,445,000 
26,978,000 

40,499,000 
35,715,000 
31,427,000 
41,867,000 
33,594,000 



April. 



lbs. raised 
one foot. 

22,982,000 

26,064,000 

32,723,000 
23,626,000 

41,888,000 
33,934,000 
33,564,000 
41,823.000 
33,932,000 



May. 



lbs. raised 
one foot. 

23,608,000 

29,032,000 
31,520,000 
29,702,000 

38,233,000 
33,714,000 
33,967,000 
40,615,000 



June. 



lbs. raised 
one foot. 

23,836,000 

30,336,000 

34,352,000 
34,846,000 

38,143,000 
34,291,000 
30,105,000 
42,098,000 
35,797,000 



These numbers are less than the immediate power of the en- 
gines, by the friction and loss of effect in working the pumps ; 
hence in comparing them with Mr. Tredgold's table, it may 
be inferred that he made his calculations from such data as can 
be realized in practice. It is known from experience, that a 
cubic foot of water can be converted into steam equal in force 
to the atmosphere, with 7 lbs. of Newcastle coals ; but we 
also know the attention necessary to produce that effect, and 
therefore have assumed that 8 T 4 ^ lbs. will be required for that 
purpose. 



STEAM-ENGINES, CANALS, RAIL-ROADS. 



375 



According to Mr. Lean's Monthly Report, for January, 1833, 
the following engines raise more than 50 millions of pounds, 
one foot high, by consuming a bushel of coals : 



Cardrew Downs - - - 
Binner Downs - - - 

Ditto 

Consolidated Mines - - 

Ditto 

Ditto 

Ditto 

Ditto 

Ditto 

Polgooth 

Pembroke 

East Crinnis - - - - 
Wheal Leisure - - - - 

Wheal Vor 

Ditto 

Ditto 

Poladras Downs - - - 
Great Work - - - * 
Wheal Towan - - - 

Ditto 

Wheal Falmouth Consols 
Wheal Darlington - - - 



Diameter of Cylinder. 

- - 66 inches, 

- - 70 - - 

- - 65 - - 

- - 90 - - 

- - 70 - - 

- - 65 - - 

- . 90 - - 

- - 90 - - 

- - 65 - - 

- - 66 - - 

- - 40 - - 

- - 76 - - 

- - 66 - - 

- - 80 - - 

- - 80 - - 

- - 53 - - 

- - 70 - - 

- * 60 - - 

- - 80 - - 

- - 80 - - 
70 - - 
80 - - 



lbs. 



single 



Strokes per ] 



51,831,751 
57,942,435 
55,931,852 
51,713,913 
61,846,133 
54,726,957 
59,978,983 
52,040,672 
65,617,011 
70,240,452 
61,170,237 
62,097,533 
53,506,372 
88,504,900 
65,471,147 
53,938,177 
56,766,668 
65,460,248 
73,159,628 
68,782,390 
54,334,137 
66,058,518 



7-34 
10-13 
8-5 
4-86 
5-93 
3-6 
4-69 
6-5 
5-28 
8.00 
6.67 
7.04 
3.58 
6.22 
6.07 
6.37 
8.78 
6.08 
4.9 
6-98 
6.22 
8.47 



Of the above, the engine of greatest operation, the first at 
Wheal Vor, raises the water 190 fathoms, at 7 lifts, drawing 
perpendicularly 160 fathoms, and the remainder diagonally. 
Main beam over the cylinder ; stroke in the cylinder 10 feet ; 
one balance-bob at the surface, and three underground. 



376 



STEAM-ENGINES, CANALS, RAIL-ROADS. 



TABLE III. — Showing the effects of a force of traction of 
100 lbs. at different velocities, on canals, rail-roads, and 
turnpike-roads.* (From Tredgold.) 



Velocity of Motion. 


Load moved by a power of 100 lbs. 


Miles 
per 

hour. 


Feet 

per 

second. 


On a Canal. 


On a level Rail-way. 


On a level 
Turnpike Road. 


Total 
'mass 
moved. 


Useful 
effect. 


Total 
mass 
moved. 


Useful} 
effect. 


Total 
mass 
moved. 


Useful 
effect. 


n 

3 

H 

4 

5 

6 

7 

8 

9 
10 
13-5 


3-66 

4-40 

5-13 

5-86 

7-33 

8-80 

10-26 

11-73 

13-20 

14-66 

19-9 


lbs. 

55,500 

38,542 

28,316 

21,680 

13,875 

9,635 

7,080 

5,420 

4,282 

3,468 

1,900 


lbs. 

39,400 

27,361 

20,100 

15,390 

9,850 

6,840 

5,026 

3,848 

3,040 

2,462 

1,350 


lbs. 

14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 


lbs. 

10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 


lbs. 

1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 


lbs. 

1,350 
1,350 
1,350 
1,350 
1,350 
1,350 
1,350 
1,350 
1,350 
1,350 
1,350 



This table is intended to exhibit the work that may be per- 
formed by the same mechanical power, at different velocities, 
on canals, rail-roads, and turnpike-roads. Ascending and de- 
scending by locks or canals, may be considered equivalent to 
the ascent and descent of inclinations on rail-roads and turnpike- 
roads. The load carried, added to the weight of the vessel or 
carriage which contains it, forms the total mass moved ; and the 
useful effect is the load. To find the effect on canals at different 
velocities, the effect of the given power at one velocity being 
known, it will be as 3 2 : 2-5 2 :: 55,500 : 38,542. The mass 
moved being very nearly inversely as the square of the velo- 
city ; at least, within certain limits. 

This table shows, that when the velocity is 5 miles per hour, 
it requires less power to obtain the same effect on a rail- 
way than on a canal ; and the lower range of figures is added 
to show the velocity at which the effect on a canal is only equal 
to that on a turnpike-road. By comparing the power and ton- 
nage of steam- vessels, it will be found that the rate of decrease 
of power by increase of velocity, is not very distant from the 

* Though the force of traction on a canal varies as the square of the velocity ; 
the mechanical power necessary to move the boat is usually reckoned to increase as 
the cube of the velocity. On a rail-road or turnpike the force of traction is con- 
stant ; but the mechanical power necessary to move the carriage increases as the 
velocity. 



STEAM-ENGINES, CANALS, RAIL-ROADS. 



377 



truth ; but we know that in a narrow canal the resistance in- 
creases in a more rapid ratio than as the square of the velocity :'* 
that is, within certain limits ; beyond them, there is a remark- 
able change in the circumstances of resistance. 

Table IV. — Showing the maximum quantity of labour a 
Horse of average strength is capable of performing, at dif- 
ferent velocities, on canals, rail-ivays, and turnpike-roads. 
(From Tredgold.) 





Duration 




Useful effect of one horse 


working one 


Velocity 


of the day's 


Force 


day, in 


tons drawn one mile. 


in miles per 
hour. 


work at the 


of traction 








preceding 


in lbs. 




On a 


On a good 




velocity. 




On a canal. 


level 


level turn- 








rail-way. 


pike-road. 


miles. 


hours. 


lbs. 


tons. 


tons. 


tons. 


2k 


HI 


83| 


520 


115 


14 


3 


8 


83| 


243 


92 


12 


3£ 


5 T 9 o 


83| 


153 


82 


10 


4 


4| 


83| 


102 


72 


9 


5 


2 T 9 o 


83| 


52 


57 


7-2 


6 


2 


83| 


30 


48 


6-0 


7 


ii 


83£ 


19 


41 


5-1 


8 


li 


83| 


12-8 


36 


4-5 


9 


9 
1 


83| 


9-0 


32 


4-0 


10 


3 

4 


831 


6-6 


28-8 


3-6 



Where horse power is employed for the higher velocities, 
the animals ought to be allowed to acquire the speed as gra- 
dually as possible at the first starting. This simple expedient 
will save the proprietors of horses much more than they are 
aware of ; and it deserves their attention to consider the best 



* According to the interesting researches of Du Buat, the resistance to the motion 
of boats even in canals may be regarded as proportional to the square of the velocity, 
or r as y 2 nearly, provided r be made to depend upon the transverse sec- 
tions of the vessel and the canal in which it moves, and a constant quantity 
k determinable by experiment. If c be the vertical section of the canal, b the 
vertical section of the immersed portion of the boat or barge ; then 

r = k-t-( — -f- 2 J. The medium of Du Buat's experiments gives k = 8*46, 

or r = 8*46 -T- ( — + 2 J ; but those experiments were not so numerous and va- 
ried as might be wished. See Principes (V Hydraulique, torn. ii. pp. 340, 342, 
&c. 

In cases connected with these, or kindred inquiries, where velocities in miles per 
hour, are to be reduced into velocities in feet per second, or the contrary ; the rules 
at p. 102 of this work will be found useful. 

2 K2 



378 STEAM-ENGINES, CANALS, RAIL-ROADS. 

mode of feeding and training horses for performing the work 
with the least injury to their animal powers. 

To compare the preceding table with practice at the higher 
velocities, it will be necessary to have the total mass moved, 
which is one-third more than the useful effect in this table. 
Now, the actual rate at which some of the quick coaches travel 
is 10 miles an hour ; the stages average about 9 miles ; and a 
coach with its load of luggage and passengers amounts to 
about 3 tons ; therefore the average day's work of 4 coach 
horses is 27 tons drawn one mile, or 6| tons drawn one mile 
by one horse. The table gives 3*6 tons, added § of 3*6 = 
4-8 tons drawn one mile for the extreme quantity of labour 
for a horse at that speed, upon a good level road ; from which 
should be deducted the loss of effect in ascending hills, heavy 
roads, &c, which will make the actual labour performed by a 
coach-horse average about double the maximum given by the 
table. The consequences are well known. 

According to Mr. Sevan's observations, the horses on the 
Grand Junction Canal draw 617 tons one mile, at the velocity 
of 2*45 miles per hour. 

According to Mr. Tredgold, if v be the maximum velocity 
of a horse, and v any other velocity, the immediate power of a 

horse is 250 v ( 1 ); and, when the weight of the vessel 

or carriage is to the weight of the load, as «: 1, we have 

the effective power ; and d being the hours 
1 -f n 
the horse works in one day, the day's work will be 

250 d v ( 1 — £) in lbg raiged ! mile? and 2 50 ( i _ *) « 

1 + n 
the force of traction in lbs. But if the force were immediately 

14*7 
applied, the value of v would be — - ; and to find the value 

when the waggons alone are moved, we have 1 : 



250 v ( 1 — -) _ 



>/ 1 + n 

14*7 14-7 

"V~d' x/dJTT~n) = V; whence the da y' s work is 

250 d v / , v y/ d (1 + n) , . , . . u 

( 1 ^ — — — '- ; which is a maximum when 

1 + n V 14-7 



STEAM-ENGINES, CANALS, RAIL-ROADS. 379 

96 



= d. Consequently, when the velocity is given, 



v 9 (1 + n) 

96 
we have ~z- t i equal the duration of the day's work in 

v 2 (l -f n) ^ 

hours : and —7- — ; — ^ = the effective day's work ; and 250 

' v(l +n)* J 

(l — J = 83£ lbs. But we may assume n to be always 

72 
so near \, as not to affect the result ; and then, — 3 = d, and 

= the day's work in lbs. or very nearly — tons raised 

one mile. This being combined with the numbers of the pre- 
ceding table, gives the effect of a horse on canals, rail-roads, and 
turnpike-roads. 

It must, however, be here added, that although the deduc- 
tions from Mr. Tredgold's valuable tables, as to the effects on 
canals, are tolerably accurate up to rates of 4 or 5 miles per 
hour, yet, when boats are moved on canals at rates of from 9 to 
12 or 14 miles per hour, the circumstances of the resistances 
undergo an essential change. The resistance, in fact, becomes 
so small, that passage-boats now travel at these high velocities ; 
and it is hence probable that rail-roads and canals will admit of 
a competition such as the supporters of rail-roads never antici- 
pated. 

I shall here briefly detail some of the facts, as they have been 
given in a letter widely circulated by Mr. W. Grahame of Glas- 
gow, in the Nautical Magazine, and other places. 

From the traffic by canal boats, which has been actually 
going on during the last two years and a half, on the Paisley 
canal, we learn this remarkable fact, that, while a speed of ten 
miles per hour has been maintained by the canal boats, the 
banks have sustained no injury whatever. The cause of injury, 
in truth, has been entirely suppressed by the velocity of the 
boat, which passes along the water without raising a ripple. 

About two years ago, measures were adopted for increas- 
ing the speed of the boats on the Paisley or Ardrossan canal. 
This canal is by no means favourable to such experiments, 
being both serpentine in its course, and narrow : it connects the 
town of Paisley with the city of Glasgow, and the village of 
Johnstone ; the distance being about twelve miles. The boats 



380 CANALS AND ROADS. 

employed on this canal are 70 feet in length, and 5*6 broad, 
and carry, if necessary, upwards of 120 passengers. They are 
formed of light iron plates, and ribs covered with wood, and 
light oiled cloth, at a whole cost of about 125/. They perform 
stages of four miles in an interval of time varying from 22 to 
25 minutes, including all stoppages, and the horses run three or 
four of these stages alternately every day. The passengers are 
under cover, or not, as they please, no difference being made in 
this particular ; and the fare is one penny per mile in the first, 
and three farthings per mile in the second cabin. 

The horses drawing the canal boat are guided by a boy, who 
rides one of them ; and, in passing under bridges at night, a 
light is shown in the bow of the boat, by which he sees his 
way, and which light is closed when the bridge is passed. In- 
termediate passengers are also accommodated to distances even 
as small as a mile ; and the facility with which the boat stops 
when relieved from the drawing force, is such as avoids all 
danger whatever. The expense of conveying a load of eight 
tons at a rate of nine or ten miles per hour, including all out- 
lay, interest, and replacement of capital, is not more than lid. 
per mile. It is also ascertained that one ton weight may be 
carried on a canal at nearly the same speed as on the railway, 
at about lid. per mile, including an allowance for interest and 
replacement of capital. 

It is also believed, that if the breadth and curvature of the 
Paisley canal admitted boats of 90 feet length, instead of 70, 
they would carry more passengers by one-half without an addi- 
tional expense, and a decrease of labour to the horses. 

The foregoing has been deduced from calculations founded 
on the observation of facts relating to the wear and tear of 
boats and horses, and the absolute resistance which these 
boats meet with in passing through the water. On this sub- 
ject it has been observed, that, in addition to the common re- 
sistance of the water to the motion of the boat, a wave, or body 
of water, is also raised before it, varying in its height accord- 
ing to the velocity of the boat, and constantly presenting an 
obstacle to her progress, provided that she only moves through 
the water at a certain slow rate. The height of this wave will 
then amount to nearly two feet, often overflowing the banks 
of the canal, and, from the obstruction it occasions, eventu- 
ally obliging the boat to be stopped. Now, if, instead of 
stopping the boat when this wave is raised, her velocity be 
increased beyond what it had then been, she advances and 
passes over it, and leaves it to subside in her wake, which it 
does, and the water becomes perfectly still. The same horses 



CANALS AND ROADS. 381 

drawing the boat at this increased speed, are found to perform 
their work better, the resistance to their progress having become 
less ; and the more the velocity of the boat is thus increased, 
the less resistance she meets with, merely having to cut the 
still water instead of the wave. It is a curious fact, that the 
wave produced by the approach of a slow canal boat is often 
observed at the distance of a mile, and upwards, along the canal, 
before the arrival there of the boat. But in the case of the 
high wave being raised by the Paisley canal boat, it is custom- 
ary to stop the boat, and after it has subsided to start again at a 
greater velocity. When the boat is to be stopped for any pur- 
pose, as her speed decreases the wave rises in proportion, and 
washes over the banks, until the motion of the boat becomes so 
small as to produce none. The discovery is doubtless a very 
important one, and, if turned to account, is likely to produce a 
material alteration in the rate of transport on canals. It was 
not known until these experiments were made, that if a boat, 
from a state of rest, was dragged along a canal, in proportion as 
her speed increased to a certain limit, that the power required 
was greater : but that, if she were started at, and preserved a 
speed exceeding the same limit, the power required would be 
less, and would decrease as her velocity increased. In fact, 
from a certain velocity there seems to be no limit to the rate at 
which a boat, as far as animal power can be applied, may thus 
pass through the water ; and as the rate increases, the power 
required decreases. On this principle it is, that the boats on 
the Paisley canal, with ninety passengers in them, are drawn by 
horses at a speed of ten miles an hour ; while it would kill 
them to draw the same boat along the canal at six miles an 
hour. A boat might indeed travel fifteen or twenty miles an 
hour easier than at six miles. The former of these velocities 
has already been attained by Mr. Grahame, along a distance 
of two miles, and is considered by him safer both for the boat 
and the canal. 

As a proof of what may be done by this method of carriage, 
Mr. Grahame states that he has performed a voyage of fifty- 
six miles along two canals in six hours and thirty-eight minutes, 
which included the descent of five, and the ascent of eleven 
locks, the passage of eighteen draw-bridges where the tracking- 
line was thrown off, and sixty common bridges, besides a tun- 
nel half a mile long ; all, of course, producing some delay. The 
boat which performed this was sixty-nine feet long, and nine 
broad, drawn by two horses, and carried thirty-three passen- 
gers, with their luggage and attendants. 

These facts furnish great encouragement to canal companies, 
to improve the construction of their boats and the speed on their 
50 2l2 



382 CANALS AND ROADS 

canals ; and thus, probably, in some positions, supersede the 
necessity of rail-roads. 

Mr. Macneill, the assistant engineer upon the Holyhead 
road, under Mr. Telford, in the course of his examination 
before a committee of the House of Commons, on steam- 
carriages, rail-roads, &c. gave the subjoined curious infor- 
mation. 

Well made roads, formed of clean hard broken stone, placed 
on a solid foundation, are little affected by changes of atmosphere; 
but weak roads, or such as are imperfectly formed with gravel, 
flint, or round pebbles, without a bottoming of stone, pavement, 
or concrete, are much affected. 

On the generality of roads, the proportional injury from the 
weather and traffic is nearly as follows. When travelled by fast 
coaches : from atmospheric changes 20; coachwheels 20; horses' 
feet 60 = 100. When travelled by wagons : atmospheric 
changes 20 ; wagon- wheels 35*5 ; horses' feet 44*5 = 100. 
Has ascertained, from a number of observations, that the wear 
of the iron tire of fast-going coachwheels is, compared with that 
of the shoes of the horses which draw them, as 326*8 to 1000, 
or as 1 to 3-4ths nearly ; and infers that the comparative injury 
done by them to roads is nearly in the same proportion. In 
the case of slow-going carriages and horses the proportion is as 
309 to 360, or as 1 to 1*16, or nearly 1 to H. The tire of the 
wheels of the fast-going coaches last from two to three months, 
according to the weather, the workmanship, and quality of iron ; 
about 20 years ago, it did not last seven days on an average. 
Coach-horse shoes remain in use about thirty days ; wagon- 
horse shoes about five weeks on an average. Where roads are 
weak, and yield under pressure, the injury caused by heavy 
wheels is far greater than on solid firm roads. 

It was found, in one instance, that the wear of hard stone, 
placed on a wet clay bottom, was four inches, while it was not 
more than half an inch when placed on a solid dry foundation. 
On the Highgate archway road, the annual wear is not more 
than half an inch in depth. To the same gentleman we owe the 
following useful table. 



CANALS AND ROADS. 



383 



Table V. — The general Results of Experiments made with 
a Stage Coach, weighing, exclusive of seven passengers, 
18 cwt., on the same piece of road, on different inclina- 
tions, and at different rates of velocity , furnish the follow- 
ing statement. 



Rate of Inclination. Rates of Travelling. Force required. 

1 in 20 6 miles per hour 268 lbs. 

1 in 26 6 213 

1 in 30 6 165 

1 in 40 6 160 



n 600 6 

n 20 8 

n 26 8 

n 30 8 

n 40 8 

n 600 8 



n 20 . . . 
n 26 . . . 
n 30 . . . 
n 40 . . . 
n 600 10 



10 
10 
10 
10 



111 

296 
219 
196 
166 
120 

318 
225 
200 
172 
128 



384 STRENGTH OP ANIMALS. 

CHAPTER XV. 

ACTIVE AND PASSIVE STRENGTH. 

Section I. — Active Strength, or Animal Energy, as of Men, 
Horses, fyc. 

1. The force obtained through the medium of animal agency 
evidently varies, not only in different species of animals, but, 
also, in different individuals. And this variation depends, first, 
on the particular constitution of the individual, and upon the 
complication of causes, which may influence it ; secondly, upon 
the particular dexterity acquired by habit. It is plain, that such 
a variation cannot be subjected to any law, and that there is no 
expedient to which we can have recourse, but that of seeking 
mean results. 

Secondly, the force varies according to the nature of the 
labour. Different muscles are brought into action in different 
gestures and positions of an animal which labours ; the weight 
itself of the animal machine is an aid in some kinds of labour, 
and a disadvantage in others : whence it is not surprising that 
the force exerted is different in different kinds of work. Thus 
the force exerted by a man is different, in carrying a weight, 
in drawing or pushing it horizontally, and in drawing or push- 
ing it vertically. 

Thirdly, the force varies according to the duration of the 
labour. The force, for example, which man can exert in an 
effort of a few instants, is different from that which he can main- 
tain equably in a course of action continued, or interrupted only 
by short intervals, for a whole day of labour, without inducing 
excessive fatigue. The former of these may be called Absolute 
Force, the latter Permanent Force. It is of use to become 
acquainted with them both, as it is often advantageous to avail 
ourselves sometimes of the one, sometimes of the other. 

Lastly, the force varies according to the different degrees of 
velocity, with which the animal, in the act of labouring, moves 
either its whole body, or that part of it which operates. The 
force of the animal is the greatest, when it stands still ; and 
becomes weaker as it moves forward, in proportion to its speed ; 
the animal acquiring, at last, such a degree of velocity as ren- 
ders it incapable of exerting any force. 

2. Let $ be a weight equivalent to the force which a man 
can exert, standing still ; and let v be the velocity with which, 



STRENGTH OF ANIMALS. 385 

if he proceeds, he is no longer capable of exerting any force : 
also, let f be a weight equivalent to the force which he exerts, 
when he proceeds, equably, with a velocity v. 

Then f will be a function of v, such that, 1st, it decreases 
whilst v increases ; 2dly, when v — 0, then f — $ ; 3dly, 
when v = v, f = 0. 

3. Upon the nature of this function, we have the three fol- 
lowing suppositions. 



1. f = $ (l ). (Bouguer, Man. des Vais.) 

2. f = t (l — —\ . (Euler, Nov. Comm. Pet. torn. III.) 

3. f = $ (l — -) (lb. torn. VIII ; and Act of Rowers.) 

4. Coroll. 1. The effect of the permanent force being mea- 
sured by the product f v, the expression for the effect will be 
one of the three following, accordingly as one or other of the 
suppositions is adopted. 



(l -Jl),o r t „(i-I) ! 



3. F V. 

5. Coroll. 2. To know the weight with which a man should 
be loaded* or the velocity with which he ought to move, in 
order to produce the greatest effect, we must make d . f v — 0. 

Whence we shall have 

1. p = -$ ; and v = h v. 

2 

2 1 

2. f = - $ ; and v — — -. v = 0-5773 v. 

3 v/3 

4 j l 

3. r = -»; and v = — v. 

9 ' 3 

6. Coroll. 3. And the value of the greatest effect will be, 
according to the 1st hypothesis . . . . = i * v : 

2 

2d = =<?> v = 0-3836 $ v : 

3 \/3 

3d - A* v ' 

2L 



386 



ANIMAL STRENGTH * SCHULZE 7 S EXPERIMENTS. 



But which of the three suppositions ought we to prefer ? 
And are we certain that any of them approximates to the true 
law of nature ? 

Mr. Schulze made a series of experiments with a view to the 
determination of this point,* and with regard to men decided 
in favour of the last of Euler's formulae : 



viz. F = $ ( 1 J 



As the experiments of this philosopher are very little known 
in England, I shall here present his brief account of them. 

7. To make the experiments on human strength, he took at 
random 20 men of different sizes and constitutions, whom he 
measured and weighed. The result is exhibited in the follow- 
ing table. 



Order. 


Height. 


Weight. 


Order. 


Height. 


Weight. 


1 


5' 3" 4'" 


122 


11 


5' 9" 7'" 


132 


2 


5 2 3 


134 


12 


5 14 


157 


3 


5 7 2 


165 


13 


5 3 2 


175 


4 


5 5 


131 


14 


5 4 1 


117 


5 


5 11 2 


177 


15 


5 10 8 


192 


6 


6 4 


158 


16 


5 3 


133 


7 


5 8 3 


180 


17 


4 11 2 


147 


8 


5 2 1 


117 


18 


5 3 9 


124 


9 


5 4 8 


140 


19 


5 6 


163 


10 


5 4 


126 


20 


5 10 1 


181 



Here the heights are expressed in feet (marked '), inches ("), 
twelfths ('"), the feet being those of Rhinland, each 12*35 Eng- 
lish inches. The weights are in pounds, which are to our 
avoirdupois lbs. as 30 to 29. 

To find the strength that each of these men might exert to 
raise a weight vertically, Mr. Schulze made the following expe- 
riments : 

He took various weights increasing by 10 lbs. from 150 lbs. 
up to 260 lbs. ; all these weights were of lead, having circular 
and equal bases. To use them with success in the proposed 
experiments, he had at the same time a kind of bench made, 
in the middle of which was a hole of the same size as the base 
of the weights : this hole was shut by a circular cover when 
pressed against the bench ; at other times it was kept 
at about the distance of a foot and a half above the bench 



* Mem. Acad. Scienc. Berlin, for 1783. 



ANIMAL STRENGTH : SCHULZE S EXPERIMENTS. 



387 



by means of a spring and some iron bars. To prevent the 
weight with which this cover was loaded during the experiment 
from forcing down the cover, lower than the level of the sur- 
face of the bench, he had several grooves made in the four iron 
bars, which sustained the cover, and which, at the same time, 
served to hold up the cover at any height where it might arrive 
by the pressure of the springs as soon as the pressure of the 
weight ceased. 

After having laid the 150 lbs. weight on the cover, and the 
other weights in succession increasing by 10 lbs. up to 250 lbs., 
he made the following experiments with the men whose size 
and weight are given above, by making them lift up the 
weights as vertically as possible all at once, and by observing 
the height to which they were able to lift them. The annexed 
table gives the heights observed for different weights marked at 
its head. 





150 


160 


170 


180 


190 


200 


210 


220 


230 


240 


250 




// /// 


// /// 


// /// 


;/ /// 


ti in 


// /// 


a 


/// 


// /// 


// /// 




1 


7 9 


6 4 


4 11 


4 4 


3 8 


2 8 


1 1 












2 


7 10 


6 6 


5 7 


4 7 


3 11 


2 5 


5 


1 


7 


3 






3 


7 9 


7 3 


6 5 


5 9 


4 11 


4 


3 


3 


8 


3 1 


1 4 




4 


8 3 


7 6 


7 2 


5 10 


5 3 


4 7 


4 


4 


7 


3 2 


1 3 




5 


12 4 


11 1 


9 7 


8 5 


7 10 


7 1 


5 10 


6 


6 


4 1 


1 




6 


14 5 


14 


13 5 


12 8 


11 5 


10 1 


8 6 


3 


8 


1 11 


2 




7 


12 11 


11 3 


10 5 


9 3 


8 1 


6 9 


5 3 


5 


1 


3 2 


1 




8 


11 9 


10 2 


9 4 


8 11 


8 1 


6 11 


5 10 












9 


9 5 


8 3 


7 1 


5 6 


4 1 


2 9 


1 3 












10 


8 1 


6 5 


4 7 


3 9 


2 5 


1 7 


4 













This table proves that the size of the men employed to raise 
the weights vertically has considerable influence on the height 
to which they brought the same weight. We find also that 
the height diminishes in a much more considerable ratio than 
the weight increases : and we may therefore conclude, that it 
is advantageous to employ large men when it becomes neces- 
sary to draw vertically from below upwards : and on the con- 
trary, it is more advantageous to employ men of a considerable 
iveight, when it is required to lift up loads by means of a pulley 
about which a cord passes, that the workmen may draw in a 
vertical direction, from above downwards. To find the absolute 
strength of these men in a horizontal direction, Mr. Schulze 
proceeded thus : 



388 



ANIMAL STRENGTH : SCHULZE's EXPERIMENTS. 



Having fixed over an open pit a brass pulley extremely well 
made, of 15 inches diameter, whose axis, made of well polished 
steel to diminish the friction, was § inch in diameter, he 
passed over this pulley a silk cord, worked with care, to give 
it both the necessary strength and flexibility. One of the ends 
of this cord carried a hook to hang a weight to it which hung 
vertically in the pit, whilst the other end was held by one of 
the 20 men, who in the first order of the following experiments 
made it pass above his shoulders ; instead of which, in the se- 
cond, he simply held it by his hands. 

Mr. S. had taken the precaution to construct this in such a 
manner that the pulley might be raised or lowered at pleasure, 
in order to keep the end of the cord held by the man always in 
a horizontal direction, according as the man was tall or short, 
and exerted his strength in any given direction. 

He had made the necessary arrangements so as to be able to 
load successively the basin of a balance which was attached to 
the hook at the end of the cord which descended into the pit, 
whilst the man who held the other end of this cord employed 
all his strength without advancing or receding a single inch. 

The following tables give the weights placed in the basin 
when the workmen were obliged to give up, having no longer 
sufficient strength to sustain the pressure occasioned by the 
weight. To proceed with certainty, Mr. S. increased the weight 
each time by five pounds, beginning from 60, and took the pre- 
caution to make this augmentation in equal intervals of time ; 
having always precisely a space of 10 seconds between them. 
The result of these observations repeated several days in succes- 
sion, is contained in the following tables. 

I. When the cord passed over the shoulders of the workmen : 



Order. 


lbs. 


Order. 


lbs. 


Order. 


lbs. 


Order. 


lbs. 


1 


95 


6 


100 


11 


95 


16 


95 


2 


105 


7 


115 


12 


100 


17 


100 


3 


110 


8 


105 


13 


110 


18 


90 


4 


100 


9 


95 


14 


90 


19 


100 


5 


105 


10 


90 


15 


110 


20 


100 



ANIMAL STRENGTH : SCHULZE's EXPERIMENTS. 389 



II. When the cord was simply held before the man : 



Order. 


lbs. 


Order. 


lbs. 


Order. 


lbs. 


Order. 


lbs. 


1 


90 


6 


100 


11 


90 


16 


90 


2 


105 


7 


110 


12 


90 


17 


90 


3 


105 


8 


100 


13 


100 


18 


85 


4 


90 


9 


90 


14 


85 


19 


100 


5 


95 


10 


85 


15 


105 


20 


100 



These two tables show that men have less power in drawing 
a cord before them than when they make it pass over their 
shoulders ; they show, also, that the largest men have not always 
the greatest strength to hold, or to draw in a horizontal direc- 
tion, by means of a cord. To obtain the absolute velocity of 
these 20 men, Mr. S. proceeded as follows : 

Having measured very exactly a distance of 12,000 Rhinland 
feet in a plain nearly level, he caused these 20 men to march with 
a fair pace, but without running, and so as to continue during 
the period of four or five hours ; the following is the time em- 
ployed in describing this space, with the velocity resulting for 
each of them. 



Order. 


Time. 


Veloc. 


Order. 


Time. 


Veloc. 


Order. 


Time. 


Veloc. 


1 


40-18 


4-94 


8 


4009 


4-99 


15 


36-17 


5-51 


2 


41-12 


4-85 


9 


40-20 


4-96 


16 


41-28 


4-82 


3 


39-08 


5-55 


10 


40-51 


4-90 


17 


42-25 


4-71 


4 


39-40 


5-04 


11 


36-17 


5-51 


18 


40-19 


4-98 


5 


34-19 


5-83 


12 


38-11 


5-24 


19 


39-57 


5-01 


6 


35-11 


5-68 


13 


38- 5 


5-25 


20 


37-51 


5-29 


7 


38- 7 


5-25 


14 


37- 1 


5-40 









It is necessary to mention, with regard to these experiments, 
that Mr. Schulze took care to place at certain distances persons 
in whom he could place confidence, in order to observe whether 
these men marched uniformly, and sufficiently quick, without 
running. 

Having thus obtained, not only the absolute force, but the 
absolute velocity also, of several men, he took the following 
method to determine their relative force. 

He made use of a machine composed of two large cylinders 

of very hard marble, which turned round a vertical cylinder of 

wood, and moved by a horse, which described in his march a 

circle of 10 Rhinland feet. This machine appeared the most 

51 2 l 2 



390 ANIMAL STRENGTH 

proper to make the subsequent experiments, which serve to 
determine the relative strength that the men had employed to 
move this machine, and which is used hereafter to determine 
which of Euler's two formulae ought to be preferred. 

To obtain this relative force, he took here the same pulley 
which served in the preceding experiments, by applying a 
cord to the vertical cylinder of wood, and attaching to the 
other end of this cord, which entered into an open pit, a suffi- 
cient weight to give successively to the machine different ve- 
locities. 

Having applied in this manner a weight of 215 lbs., the 
machine acquired a motion, which after being reduced to an 
uniform velocity, taking into account the acceleration of the 
weight, of the friction, and of the stiffness of the cord, gave 2*41 
feet velocity ; and having applied in the same manner a weight 
of 220 lbs., the resulting uniform motion gave a velocity of 2*47 
feet. These two limits are mentioned because they serve as a 
comparison with what immediately follows : Mr. S. began these 
experiments with a weight of 100 lbs., and increased it by 5 
every time from that number up to 400 lbs. 

He made this machine move by the seven first of his work- 
men, placing them in such a way that their direction remained 
almost always perpendicular to the arm on which was attached 
the cord which passed over their shoulders in an almost hori- 
zontal direction. 

Thus situated, they made 281 turns with this machine in two 
hours, which gave for their relative velocity v = 2*45 feet per 
second. We have also the absolute force, or $>, from these 7 men, 
by the above table, = 730 lbs. : and their absolute velocity or 
v as 5-30 feet. 

Therefore, by substituting these values in the first formula, 
we find the relative force f = 205 lbs., which agrees very well 
with what we have just found above. 

If instead of this first formula the second be taken, it gives 
f = 153 lbs., which is far too little. 

By this it is evident, that the last of Euler's two formulae is 
to be preferred in all respects. Mr. Schulze made a great 
number of combinations, and almost always found the same 
effect. 

Dividing the 205 lbs. which we have just found by 7, the 
number of workmen, we get 29 lbs. for the relative force 
with 2-45 feet relative velocity for each man, which is rather 
more than the values commonly adopted in the computation 
of machinery. A number of other observations on different 
machines have given the same result ; that is to say, we 



ANIMAL STRENGTH : MEN. 391 

must value the mean human strength at 29 or 30 lbs. with a 
velocity of 2\ feet per second. 

To obtain the ratio of the strength of a horse to that of a man, 
Mr. Schulze proceeded in a similar manner ; but his results, 
in reference to that inquiry, are neither so correct nor so inter- 
esting. 

8. In the first volume of my Mechanics, I stated the average 
force of a man at rest to be 70 lbs., and his utmost walking ve- 
locity when unloaded to be about 6 feet per second ; and thence 
inferred that a man would produce the greatest momentum 
when drawing 31|lbs. along a horizontal plane with a velocity 
of 2 feet per second. But that is not the most advantageous 
way of applying human strength. 

9. Dr. Desaguliers asserts, that a man can raise of water or 
any other weight about 550 lbs., or one hogshead (weight of the 
vessel included), 10 feet high in a minute ; this statement, 
though he says it will hold good for 6 hours, appears from his 
own facts to be too high ; and is certainly such as could not be 
continued one day after another. Mr. Smeaton considers this 
work as the effect of haste or distress ; and reports that 6 good 
English labourers will be required to raise 21141 solid feet of 
sea water to the height of four feet in four hours : in this case 
the men will raise a very little more than 6 cubic feet of fresh 
water each to the height of 10 feet in a minute. Now the hogs- 
head containing about 8§ cubic feet, Smeaton's allowance of 
work proves less than that of Desaguliers in the ratio of 6 to 8§ 
or 3 to 4%. And as his good English labourers who can work 
at this rate are estimated by him to be equal to a double set of 
common men picked up at random, it seems proper to state 
that, with the probabilities of voluntary interruption, and other 
incidents, a man's work for several successive days ought not to 
be valued at more than half a hogshead raised 10 feet high in a 
minute. Smeaton likewise states that two ordinary horses will 
do the work in three hours and twenty minutes, which amounts 
to little more than two hogsheads and a half raised 10 feet high 
in a minute. So that, if these statements be accurate, one horse 
will do the work of five men. 

Mr. Emerson affirms, that a man of ordinary strength, turn- 
ing a roller by the handle, can act for a whole day against a 
resistance equal to 30 lbs. weight ; and if he works 10 hours a 
day, he will raise a weight of 30 lbs. through 3% feet in a second 
of time ; or, if the weight be greater, he will raise it to a pro- 
portionally less height. If two men work at a windlass or 
roller, they can more easily draw up 70 lbs. than one man can 
30 lbs. ; provided the elbow of one of the handles be at right 



392 ANIMAL STRENGTH : MEN. 

angles to that of the other. Men used to bear loads, such as 
porters, will carry from 150 lbs. to 200 or 250 lbs. according to 
their strength. A man cannot well draw more than 70 lbs. or 
80 lbs. horizontally : and he cannot thrust with a greater force 
acting horizontally at the height of his shoulders than 27 or 
30 lbs. But one of the most advantageous ways in which a 
man can exert his force is to sit and pull towards him nearly 
horizontally, as in the action of rowing. 

M. Coulomb communicated to the French National Institute 
the results of various experiments on the quantity of action 
which men can afford by their daily work, according to the dif- 
ferent manners in which they employ their strength. In the 
first place he examined the quantity of action which men can 
produce when, during a day, they mount a set of steps or stairs, 
either with or without a burthen. He found that the quantity 
of action of a man who mounts without a burthen, having only 
his own body to raise, is double that of a man loaded with a 
weight of 68 kilogrammes, or 150 lbs. avoirdupois, both con- 
tinuing at work for a day. Hence it appears how much, with 
equal fatigue and time, the total or absolute effort may obtain 
different values by varying the combinations of effort and 
velocity. 

But the word effect here denotes the total quantity of labour 
employed to raise, not only the burthen, but the man himself ; 
and as Coulomb observes, what is of the greatest importance to 
consider is the useful effect, that is to say, the total effect, de- 
ducting the value which represents the transference of the 
weight of the man's body. This total effect is the greatest pos- 
sible when the man ascends without a burthen ; but the useful 
effect is then nothing : it is also nothing if the man be so much 
loaded as to be scarcely capable of moving : and consequently 
there exists between these two limits a value of the load such 
that the useful effect is a maximum. M. Coulomb supposes that 
the loss of quantity of action is proportional to the load (an 
hypothesis which experience confirms), whence he obtains an 
equation which, treated according to the rules of maxima and 
minima, gives 53 kilogrammes (117 lbs. avoird.) for the weight 
with which the man ought to be loaded, in order to produce 
during one day, by ascending stairs, the greatest useful effect : 
the quantity of action which results from this determination has 
for its value 56 kilogrammes (123^ lbs. avoird.) raised through 
one kilometer, or nearly 1094 yards. But this method of work- 
ing is attended with a loss of three-fourths of the total action of 
men, and consequently costs four times as much as work, in 
which, after having mounted a set of steps without any burthen, 



ANIMAL STRENGTH : MEN. 393 

the man should suffer himself to fall by any means, so as to 
raise a weight nearly equal to that of his own body. 

From an examination of the work of men walking on a hori- 
zontal path, with or without a load, M. Coulomb concludes that 
the greatest quantity of action takes place when the men walk 
being loaded ; and is to that of men walking under a load of 
58 kilogrammes (128 lbs. avoird.) nearly as 7 to 4. The weight 
which a man ought to carry in order to produce the greatest 
useful effect, namely, that effect in which the quantity of action 
relative to the carrying his own weight is deducted from the 
total effect, is 50*4 kilogrammes, or 111*18 lbs. avoirdupois. 

There is a particular case which always obtains with respect 
to burthens carried in towns, viz. that in which the men, after 
having carried their load, return unloaded for a new burthen. 
The weight they should carry in this case, to produce the great- 
est effect, is 61*25 kilogrammes (135|lbs. avoird.) The quan- 
tity of useful action in this case compared with that of a man 
who walks freely and without a load, is nearly as 1 to 5, or, in 
other words, he employs to pure loss f of his power. By 
causing a man to mount a set of steps freely and without 
burthen, his quantity of action is at least double of what he 
affords in any other method of employing his strength. 

When men labour in cultivating the ground, the whole 
quantity afforded by one during a day amounts to 100 kilo- 
grammes elevated to one kilometer, that is, 220*6 lbs. raised 
1094 yards. M. Coulomb, comparing this work with that of 
men employed to carry burthens up an ascent of steps, or at 
the pile-engine, finds a loss of about ¥ V part only of the quan- 
tity of action, which may be neglected in researches of this 
kind. 

In estimating mean results we should not determine from 
experiments of short duration, nor should we make any deduc- 
tions from the exertions of men of more than ordinary strength. 
The mean results have likewise a relation to climate. " I have 
caused," says M. Coulomb, " extensive works to be executed 
by the troops at Martinico, where the thermometer (of Reau- 
mur) is seldom lower than 20° (77° of Fahrenheit). I have 
executed works of the same kind by the troops in France ; 
and I can affirm that under the fourteenth degree of latitude, 
where men are almost always covered with perspiration, they 
are not capable of performing half the work they could perform 
in our climate." 

10. Entirely according with these are the experiments of 
Regnier, by means of a dynamometer, the results of which not 
only established the superiority of civilized men over savages, 



394 ANIMAL STRENGTH : MEN. 

but that of the Englishman over the Frenchman. The fol- 
lowing is reduced from one of Regnier's tables of mean 
results. 



STRENGTH. 



Savages, of Van Dieman's Land . 

New Holland . . . . 

Timor 

Civilized men : French 

English 



With the 
hands. 



lbs. oz. 

30 6 

51 8 

58 7 

69 2 

71 4 



With the 
reins. 



lbs* oz. 



14 8 

16 2 

22 1 

23 8 



11. A porter in London is accustomed to carry a burthen of 
200 lbs. at the rate of three miles an hour : and a couple of 
chairmen continue at the rate of four miles an hour, under a 
load of 300 lbs. Yet these exertions, Professor Leslie remarks, 
are greatly inferior to the subsultory labour performed by 
porters in Turkey, the Levant, and generally on the shores of 
the Mediterranean. At Constantinople, an Albanian porter 
will carry 800 or 900 lbs. on his back, stooping forward, and 
assisting his steps by a short staff. Such loads, however, are 
carried for very short intervals. At Marseilles it is affirmed 
that four porters carry the immense load of nearly two tons, by 
means of soft hods passing over their heads, and resting on their 
shoulders, with the ends of poles from which the goods are sus- 
pended. 

12. With regard to the magnitude of the comparative efforts 
of man in different employments, the late Mr. Robertson Bu- 
chanan ascertained, that in working a pump, in turning a winch, 
in ringing a bell, and rowing a boat, the dynamic results are as 
the numbers 100, 167, 227, and 248. 

According to the interesting experiments described in M. 
Hachette's Traite des Machines, the dynamic unit being the 
weight of a cubic metre of water raised to the height of 
one metre [that is, 2208 lbs. avoird. or 4 hogsheads raised to 
the height of 3*281 feet, or 1*3124 hogsheads to the height of 
10 feet], we have the following measures, at a medium, of the 
daily actions of men. 

Dyn. Unit. 

1. A man marching 1\ hours on a slope of 7 degrees, 

with a load of from 15 to 18 lbs 225 

2. Marching in a mountainous country without load. 140 



ANIMAL STRENGTH : MEN. 395 

3. Carrier of wood up a ladder, his weight 123, his 

load 117 lbs 109 

4. Carrier of peat up steps, his own weight com- 

prised, 112 to 120 

5. Man working at the cord of a pulley to raise the / 

ram of a pile engine : three examples . . . C 

6. A man drawing water from a well by means of a 

cord 71 

7. Man working at a capstan 116 

8. Man working at a capstan to raise water, mean 

of 24 110 

The unit of transport being the weight of a cubic metre of 
water, carried a metre (2208 lbs. 3281 feet) upon a horizontal 
road, we have for the daily action. 

Dyn. Unit. 

1. A man travelling without load on a flat road, his 

weight 154 lbs. his journey 3H miles .... 3500 

2. A soldier, carrying from 44 to 55 lbs. travelling 12£ 

miles, 1800 to 1900 

3. Ditto a forced march of 25 miles 2800 

4. A French porter, weight of the man not in- 

cluded, 792 to 800 

5. Porter with wheel-barrow, weight of the man not 

included 1015 

6. Porters with a sledge 627 

7. A man drawing a boat on a canal; 110310 lbs. 

conveyed 6| miles 550000 

14. Mr. B. Bevan, an able engineer, has made experiments 
on the application of human energy to the use of augurs, gim- 
blets, screw-drivers, &c. He has presented to the public the 
following list, as a specimen ; premising that many ordinary 
operations are performed in a short space of time, and may, 
therefore, be done by greater exertion than if a longer time 
was necessary. Thus a person, for a short time, is able to use a 
tool or instrument called 

lbs. 

A drawing-knife, with a force of . ... 100 

An augur, with two hands 100 

A screw-driver, one hand 84 

A common bench vice handle 72 

A chisel and awl, vertical pressure ... 72 

A windlass, handle revolving 60 



396 ANIMAL STRENGTH : MEN. 

lbs. 

Pincers and pliers, compression .... 60 

A hand-plane, horizontally 50 

A hand or thumb-vice 45 

A hand-saw 36 

A stock-bit, revolving 16 

Small screw-drivers, or twisting by the 

thumb and fore-finger only 14 

15. M. Morisot informs us that the time employed by a 
French stone-mason's sawyer, to make a section of a square 
toise (40-89 sq. feet Eng.) in different stones, is as below : 
viz. 

hours. 

Calcareous stone, equal grain, spec. grav. 2200 . . 45 

hard, spec. grav. 2300 62 

Liais, ditto hard, fine grain, spec. grav. 2400. ... 67 

Pyrenean alabaster, the softest of the marbles ... 56 

Normandy granite 504 

Granite from Vosges . 700 

Red and green porphyry 1177 

The workmen ordinarily made 50 oscillations in a minute ; 
each stroke about 15| inches. 

16. Hassenfratz assigns 1 3 kilogrammes as the mean effort of 
such a man ; but M. Navier, in his new edition of Belidor, 
Architecture Hydraulique, regards this estimate as too high. 
If this were correct, the daily quantity of action of the sawyer 
would be equivalent to 376 kilogrammes elevated to a kilo- 
metre (818 lbs. raised f of a mile), a quantity more than triple 
that of a man working at a winch. M. Navier gives, as a more 
correct measure of this labour for 12 hours, 188 kilomgrammes 
raised a kilometre : half the former measure. But all this is 
probably very vague. 

17. Among quadrupeds, those which are employed to pro- 
duce a mechanical effect are, the dog, the ass, the mule, the 
ox, the camel, and the horse. Of these the horse is the only 
one, so far as we are aware, whose animal energy has been 
subjected to cautious experiments ; and even with regard to 
this noble animal, opinions as to actual results are very much 
afloat. The dynamic effort of the horse is, however, pro- 
bably, about six times that of a strong and active labourer. 
Desaguliers states the proportion as 5 to 1, coinciding with 
the deductions of Smeaton. The French authors usually 
regard seven men as equivalent to one horse. As a fair mean 
between these, I assumed in vol. I. of my Mechanics the pro- 
portion of 6 to 1 and stated the strength of a horse as equiva- 



ANIMAL STRENGTH I HORSES. 397 

lent to 420 lbs. at a dead pull. But the proportion must not 
be regarded as constant ; but obviously varies much according 
to the breed and training of the animal, as well as according to 
the nature of the work about which he is employed. Thus the 
worst way, as De la Hire observed, of applying the strength of 
a horse is to make him carry a weight up a steep hill ; while 
the organization of a man fits him very well for that kind of la- 
bour : hence three men climbing up such a hill with a weight 
of 100 lbs. each, will proceed faster than a horse with a load 
of 300 lbs. 

18. In the memoirs of the French Academy for 1703 are in- 
serted the comparative observations of M. Amontons, on the 
velocity of men and of horses ; in which he states the velocity 
of a horse loaded with a man and walking to be rather more 
than 5i feet per second, or 3^ miles per hour, and when going 
a moderate trot with the same weight to be about 85 feet per 
second, or about 6 miles per hour. These velocities, however, 
are somewhat less than what might have been taken for the 
mean velocities. 

19. But the best way of applying the strength of horses is 
to make them draw weights in carriages, &c. To this kind 
of labour, therefore, the inquiries of experimentalists should 
be directed. A horse put into harness and making an effort 
to draw, bends himself forward, inclines his legs, and brings 
his breast nearer to the earth ; and this so much the more as 
the effort is the more considerable. So that when a horse is 
employed in drawing, his effort will depend, in some measure, 
both upon his own weight and that which he carries on his back. 

Indeed it is highly useful to load the back of a drawing horse 
to a certain' extent ; though this, on a slight consideration, 
might be thought to augment unnecessarily the fatigue of the 
animal ; but it must be recollected that the mass with which 
the horse is charged vertically is added in part to the effort 
which he makes in the direction of traction, and thus dispenses 
with the necessity of his inclining so much forward as he must 
otherwise do : and may, therefore, under this point of view, 
relieve the draught more than to compensate for the additional 
fatigue occasioned by the vertical pressure. Carmen and wa- 
goners in general are well aware of this, and are commonly 
very careful to dispose of the load in such a manner that the 
shafts shall throw a due proportion of the weight on the back 
of the shaft horse. This is most efficaciously accomplished at 
Yarmouth, in Norfolk, where a number of narrow streets con- 
necting the market-place with the quay, have led to the inven- 
tion and use of the low, strong, narrow carts, thence denomi- 
52 2 M 



398 ANIMAL STRENGTH I HORSES. 

nated Yarmouth carts, drawn by one horse ; and on which 
the loads are frequently shifted, especially when the vehicles 
pass over the bridge, in order to give the animals better foot 
hold, and consequently a greater dynamic effort. 

20. The best disposition of the traces during the time a horse 
is drawing is perpendicular to the position of the collar upon 
his breast and shoulders : when the horse stands at ease, this 
position of the traces is rather inclined upwards from the direc- 
tion of the road ; but when he leans forward to draw the load, 
the traces should then become nearly parallel to the plane over 
which the carriage is to be drawn ; or, if he be employed in 
drawing a sledge, or any thing without wheels, the inclination 
of the traces to the road should (from the table at p. 243) be 
about I83 , when the friction is one-third of the pressure. If 
the relation of the friction to the pressure be different from this, 
the same table will exhibit the angle which the traces must 
make with the road. 

21. When a horse is made to move in a circular path, as is 
often practised in mills and other machines moved by horses, it 
will be necessary to give the circles which the animal has to 
walk round the greatest diameter that will comport with the 
local and other conditions to which the motion must be sub- 
jected. It is obvious, indeed, that since a rectilinear motion is 
the most easy for the horse, the less the line in which he moves 
is curved, with the greater facility he will walk over it, and the 
less he need recline from a vertical position : and besides this, 
with equal velocity the centrifugal force will be less in the 
greatest circle, which will proportionally diminish the friction 
of the cylindrical part of the trunnions, and the labour of mov- 
ing the machine. And further, the greater the diameter of the 
horse-walk, the nearer the chord of the circle in which the 
horse draws is to coincidence with the tangent, which is the 
most advantageous position of the line of traction. On these 
accounts it is that, although a horse may draw in a circular 
walk of 18 feet diameter, yet in general it is advisable that the 
diameter of such a walk should not be less than 25 or 30 feet j 
and in many instances 40 feet would be preferable to either. 

22. It has been stated by Desaguliers and some others, that a 
horse employed daily in drawing nearly horizontally can move, 
during eight hours in the day, about 200 lbs. at the rate of 2§ 
miles per hour, or 3| feet per second. If the weight be aug- 
mented to about 240 or 250 lbs., the horse cannot work more 
than six hours a day, and that with a less velocity. And, in 
both cases, if he carry some weight, he will draw better than 
if he carried none. M. Sauveur estimates the mean effort of a 



ANIMAL STRENGTH I HORSES. 399 

horse at 175 French, or 189 avoird. pounds, with a velocity of 
rather more than three feet per second. But all these are pro- 
bably too high to be continued for eight hours, day after day. 
In another place Desaguliers states the mean work of a horse 
as equivalent to the raising a hogshead full of water (or 550 lbs.) 
50 feet high in a minute. But Mr. Smeaton, to whose autho- 
rity much is due, asserts, from a number of experiments, that 
the greatest effect is the raising 550 lbs. forty feet high in a 
minute. And, from some experiments made by the Society for 
the Encouragement of Arts, under the direction of Mr. Samuel 
Moore, it was concluded, that a horse, moving at the rate of 
three miles an hour, can exert a force of 80 lbs. Unluckily, 
we are not sufficiently acquainted with the nature of the experi- 
ments and observations from which these deductions were made, 
to institute an accurate comparison of their results. Neither 
of them ought to express what a horse can draw upon a car- 
riage ; because in that case friction only is to be overcome (after 
the load is once put into motion) ; so that a middling horse, 
well applied to a cart, will often draw much more than 1000 
lbs. The proper estimate would be that which measures the 
weight that a horse would draw up out of a well ; the animal 
acting by a horizontal line of traction turned into the vertical 
direction by a simple pulley, or roller, whose friction should be 
reduced as much as possible. 

23. Mr. Tredgold, in his valuable publication on Rail-roads, 
has directed his attention to the subject of " horsepower." The 
following is his expression for the power of a horse, 250 

250 d v (l— —\ 

v(l ); and — ■ for the day's work in lbs. 

\ v /' 1 + n J 

raised one mile ; d being the hours which the horse works in a 
day, and the weight of the carriage to that of the load as n : 1 . 

14-7 
He also gives — — - for the greatest speed in miles per hour, 

when the horse is unloaded. These expressions must, at pre- 
sent, be regarded as tentative. The following is his tablet of 
the comparison of the duration of a horse's daily labour and 
maximum of velocity, unloaded. 



400 ANIMAL STRENGTH I HORSES. 

Duration of labour. Max. velocity unloaded 

Hours. in miles per hour. 

1 - - - - - - - 14-7 

2 10*4 

3 8-5 

4 7*3 

5 6-6 

6 - - 6-0 

7 - - - - - - - 55 

8 52 

9 4*9 

10 4*6 

Taking the hours of labour at 6 per diem, the utmost that 
Mr. Tredgold would recommend, the maximum of useful effect 
he assigns at 125 lbs. moving at the rate of three miles per 
hour, and regarding the expense of carriage, in that case, as 
unity ; then, — 



Miles per hour. 


Proportional expense. Moving force. 


2 


U or 1-125 - - - - 166 lbs. 


3 


1 .... 125 — 


Si 


lj\ or 1-0285 - - - - 104 — 


4 


U or 1-125 - - - - 83 — 


4§ 


H or 1-333 - - - - 622 — 


5 


If or 1-8 --- - 41f — 


5ft 


2 - . - . 36£ — 



That is, the expense of conveying goods at 3 miles per hour 
being 1 ; the expense at A\ miles per hour, will be U ; and 
so on, the expense being doubled when the speed is 5§ miles 
per hour. 

24. Thus, according to Mr. Tredgold, we have for the day 
of 6 hours 2250 lbs. raised one mile. And Mr. Bevan, who has 
made many experiments on the force of traction to move canal 
boats on the Grand Junction Canal, found the force of traction 
80 lbs., and the space travelled in a day 26 miles ; hence, it is 
only equivalent to 26 x 80 = 2080 lbs. raised one mile for the 
day's work ; the rate of travelling being 2-45 miles per hour ; 
and the result a little less than Mr. Tredgold's, the difference 
probably arising from the deviation of the angle of the cate- 
nary from the horizon. 

25. The following experimental data from Mr. Bevan also 
deserve attention. 

" In the period from 1803 to 1809, I had the opportunity 
of ascertaining correctly the mean force exerted by good 
horses in drawing a plough ; having had the superintendence 



ANIMAL STRENGTH I HORSES. 401 

of the experiments on that head at the various ploughing matches 
both at Woburn and Ashridge, under the patronage of the Duke 
of Bedford and the Earl of Bridgewater. I find among my 
memoranda the result of eight ploughing matches, at which 
there were seldom fewer than seven teams as competitors for 
the various prizes. 

lbs. 

The first result is from the mean force of each horse 
in six teams, of two horses each team, upon 
light sandy soil = 156 

The second result is from seven teams of two horses 
each team, upon loamy ground, near Great Berk- 
hampstead --------- 154 

The third result is from six teams of four horses each 

team, with old Hertfordshire ploughs - - = 127 

The fourth result is from seven teams of four horses 
each team upon strong stony land (improved 
ploughs) -- = 167 

The fifth result is from seven teams of four horses 
each team, upon strong stony land (old Hertford- 
shire ploughs) - - - - - - = 193 

The sixth result is from seven teams of two horses 

each team, upon light loam - - = 177 

The seventh result is from five teams of two horses 

each, upon light, sandy land - - - - = 170 

The eighth result is from seven teams of two horses 

each team, upon sandy land - - - - = 160 

The mean force exerted by each horse from fifty-two 
teams, or one hundred and forty-four horses, =163 pounds 
each horse, and although the speed was not particularly 
entered, it could not be less than the rate of two miles and a 
half an hour. 

" As these experiments were fairly made, and by horses of 
the common breed used by farmers, and upon ploughs of va- 
rious counties, these numbers may be considered as a pretty 
accurate measure of the force actually exerted by horses at 
plough, and which they are able to do without injury for many 
weeks ; but it should be remembered, that if these horses had 
been put out of their usual walking pace, the result would have 
been very different. The mean power of the draught-horse, 
deduced from the above-mentioned experiments, exceeds the 
calculated power from the highest formula of Mr. Leslie, which 
is as follows : (15 — v) 2 = lbs. avoirdupois for the traction of 
a strong horse, and (12 — v)* = lbs. traction of the ordinary 
horse, v = velocity in miles per hour." 

2 m 2 



402 PASSIVE STRENGTH. 

Section II. — Passive Strength. 

1. When a weight is supported by a bar resting on two ful- 
crums, the pressure on each is inversely as its distance from the 
weight. 

2. The strain on a given point of a bar, placed horizontally, 
and supported at both ends, from a weight placed on it, is pro- 
portional to the rectangle of the segments into which the point 
divides the bar. 

3. Hence that strain is greatest in the middle of the bar or 
beam ; or, in other words, if the bar be prismatic, it is most 
likely to break in the middle, or it is weakest there. 

4. The strain produced by the weight of an equable bar, at 
any point of its length, is equal to the strain produced by half 
the weight of one segment acting at the end of a lever equal to 
the other segment. 

5. Def. A substance perfectly elastic is initially extended 
and compressed in equal degrees by equal forces, and proportion- 
ally by proportional forces. 

6. Def. The modulus of the elasticity of any substance is 
a column of the same substance, capable of producing a pressure 
on its base which is to the weight causing a certain degree of 
compression, as the length of the substance is to the diminution 
of its length. 

The modulus of elasticity is the measure of the elastic force 
of any substance. 

A practical notion of the modulus of elasticity may be rea- 
dily obtained. Let s be the quantity of a bar of wood, iron or 
other substance, an inch square and a foot in length would be 
extended or diminished by the force/; and let / be any other 
length of a bar of equal base and like substance ; then 

1 : / : : £ : A> or / « = A? the extension or diminution in the 
length /. 

The modulus of elasticity is found by this analogy : as the 
diminution of the length of any substance is to its length, so is 
the force that produced that diminution to the modulus of elas- 
ticity. Or, denoting the weight of the modulus in lbs. for a 
base of an inch square by m ; it is 

s :f::l : m =£- 

E 

And if w be the weight of a bar of the substance one inch square 
and 1 foot in length ; then, if m be the height of the modulus 
of elasticity in feet, we have 

/- = M. 

W s 



PASSIVE STRENGTH I MODULUS OF ELASTICITY. 403 

7. When a force is applied to an elastic column of a rectangu- 
lar prismatic form in a direction parallel to the axis, the parts 
nearest to the line of direction of the force exert a resistance in 
an opposite direction ; those particles which are at a distance 
beyond the axis equal to a third proportional to the depth, and 
twelve times the distance of the line of direction of the force, 
remain in their natural state ; and the parts beyond them act in 
the direction of the force. 

8. The weight of the modulus of the elasticity of a column 
being m, a weight bending it in any manner f, the distance of 
the line of its application from any point of the axis d, and the 
depth of the column, d, the radius of curvature will be 

d 2 m 

\ztf' 

9. The distance of the point of greatest curvature of a pris- 
matic beam, from the line of direction of the force, is twice the 
versed sine of that arc of the circle of greatest curvature, of 
which the extremity is parallel to that of the beam. 

When the force is longitudinal, and the curvature inconsider- 
able, the form coincides with the harmonic curve, the curvature 
being proportional to the distance from the axis ; and the dis- 
tance of the point of indifference from the axis becomes the 
secant of an arc proportional to the distance from the middle of 
the column. 

10. If a beam is naturally of the form which a prismatic beam 
would acquire, if it were slightly bent by a longitudinal force, 
calling its depth, d, its length, /, the circumference of a circle 
of which the diameter is unity, c, the weight of the modulus of 
elasticity, m, the natural deviation from the rectilinear form, A ', 
and a force applied at the extremity of the axis, f, the total de- 
viation from the rectilinear form will be 

, _ d 2 c* Am 
A ~ d 2 c 2 m — 12 l 2 f ' 
Scholium. It appears from this formula, that when the 
other quantities remain unaltered, A' varies in proportion to 
A, and if A = o, the beam cannot be retained in a state of in- 
flection, while the denominator of the fraction remains a finite 
quantity ; but when d 2 c 2 m — 12 P f, A' becomes infinite, 
whatever may be the magnitude of A, and the force will over- 
power the beam, or will at least cause it to bend so much as to 
derange the operation of the forces concerned. In this case/= 

771 d 2 

— , .8225 — m, which is the force capable of holding 
I ' 12 I 

the beam in equilibrium in any inconsiderable degree of cur- 



(t) 



404 STRENGTH OF ELASTIC SUBSTANCES. 

vature. Hence the modulus being known for any substance, we 
may determine at once the weight which a given bar nearly 
straight is capable of supporting. For instance, in firwood, sup- 
posing its height 10,000,000 feet, a bar an inch square and ten 
feet long may begin to bend with the weight of a bar of the same 

thickness, equal in length to .8225 Xtttt; r^; X 10,000,000 

^ & 120 X 120 ? 7 

feet, or 571 feet ; that is, with a weight of about 120 lbs. ; neg- 
lecting the effect of the weight of the bar itself. In the same 
manner the strength of a bar of any other substance may be deter- 
mined, either from direct experiments on its flexure, or from the 

m I 2 I 

sounds that it produces. If/= — , -=- = .8225 n, and- = 

n a 2 a 

y/ (.8225 n) = .907 s/ n ; whence, if we know the force re- 
quired to crush a bar or column, we may calculate what must 
be the proportion of its length to its depth, in order that it may 
begin to bend rather than be crushed. 

11. When a longitudinal force is applied to the extremities 
of a straight prismatic beam, at the distance d from the axis, 
the deflection of the middle of the beam will be d . (sec. arc 

M¥) •*)->■ 

12. The form of an elastic bar, fixed at one end, and bearing 
a weight at the extremity, becomes ultimately a cubic parabola, 
and the depression is f of the versed sine of an equal arc, in the 
smaller circle of curvature. 

13. The weight of the modulus of the elasticity of a bar is to 
a weight acting at its extremity only, as four times the cube of 
the length to the product of the square of the depth and the de- 
pression. 

14. If an equable bar be fixed horizontally at one end, and 
bent by its own weight, the depression at the extremity will be 
half the versed sine of an equal arc in the circle of curvature at 
the fixed point. 

15. The height of the modulus of the elasticity of a bar, fixed 
at one end, and depressed by its own weight, is half as much, 
more as the fourth power of the length divided by the product 
of the square of the depth and the depression. 

16. The depression of the middle of a bar supported at 
both ends, produced by its own weight, is five-sixths of the 
versed sine of half the equal arc in the circle of least cur- 
vature. 

17. The height of the modulus of the elasticity of a bar, sup- 
ported at both ends, is ~ 2 of the fourth power of the length, 



STRENGTH OP ELASTIC SUBSTANCES. 405 

divided by the product of the depression and the square of the 
depth. 

From an experiment made by Mr. Leslie on a bar in these 
circumstances, the height of the modulus of the elasticity of 
deal appears to be about 9,328,000 feet. Chladni's observa- 
tions on the sounds of fir wood afford very nearly the same 
result. 

18. The weight under which a vertical bar not fixed at the 
end may begin to bend, is to any weight laid on the middle of 
the same bar, when supported at the extremities in a horizontal 
position, nearly in the ratio of T |<nro °f tne length to the de- 
pression. 

19. Def. The stiffness of bodies is measured by their re- 
sistance at an equal linear deviation from their natural po- 
sition. 

20. The stiffness of a beam is directly as its breadth, and as 
the cube of its depth, and inversely as the cube of its length. 

21. The direct cohesive or repulsive strength of a body is in 
the joint ratio of its primitive elasticity, of its toughness, and 
the magnitude of its section. 

Though most natural substances appear in their intimate con- 
stitution to be perfectly elastic, yet it often happens that their 
toughness with respect to extension and compression differs 
very materially. In general, bodies are said to have less tough- 
ness in resisting extension than compression. 

22. The transverse strength of a beam is directly as the 
breadth and as the square of the depth, and inversely as the 
length. 

Scholium. If one of the surfaces of a beam were incom- 
pressible, and the cohesive force of all its strata collected in 
the other, its strength would be six times as great as in the 

d 2 7Yl 

natural state ; for the radius of curvature would be — -, 

d/ 
which could not be less than twice as great as in the natural 
state, because the strata would be twice as much extended, with 
the same curvature, as when the neutral point is in the axis ; 
and /would then be six times as great. 

23. Def. The resilience of a beam may be considered as 
proportional to the height from which a given body must fall 
to break it. 

24. The resilience of prismatic beams is simply as their 
bulk. 

25. The stiffest beam that can be cut out of a given cylinder 
is that of which the depth is to the breadth as the square root 
of 3 to 1, and the strongest as the square root of 2 to 1 ; but the 

53 



406 STRENGTH OF ELASTIC SUBSTANCES. 

most resilient will be that which has its depth and breadth 
equal. 

26. Supposing a tube of evanescent thickness to be expanded 
into a similar tube of greater diameter, but of equal length, the 
quantity of matter remaining the same, the strength will be in- 
creased in the ratio of the diameter, and the stiffness in the ratio 
of the square of the diameter, but the resilience will remain 
unaltered. 

27. The stiffness of a cylinder is to that of its circumscribing 
prism as three times the bulk of the cylinder to four times that 
of the prism. 

28. If a column, subjected to a longitudinal force, be cut out 
of a plank or slab of equable depth, in order that the extension 
and compression of the surfaces may be initially everywhere 
equal, its outline must be a circular arc. 

29. If a column be cut out of a plank of equable breadth, and 
the outline limiting its depth be composed of two triangles, 
joined at their bases, the tension of the surfaces produced by a 
longitudinal force will be everywhere equal, when the ra- 
dius of curvature at the middle becomes equal to half the 
length of the column; and in this case the curve will be a cycloid. 

When the curvature at the middle differs from that of the 
cycloid, the figure of the column becomes of more difficult in- 
vestigation. It may, however, be delineated mechanically, 
making both the depth of the column and its radius of curvature 
proportional always to s/ a. If the breadth of the column vary 
in the same proportion as the depth, they must both be every- 
where as the cube root of a, the ordinate. — ( Young's Nat. Phil, 
vol. ii.) 

30. The modulus of elasticity has not yet been ascertained in 
reference to so many subjects as could be wished. Professor 
Leslie exhibits several, however, as below. That of white 
marble is 2,150,000 feet, or a weight of 2,520,000 pounds avoir- 
dupois on the square inch ; while that of Portland stone is only 
1,570,000 feet, corresponding on the square inch to the weight 
of 1,530,000 pounds. 

White marble and Portland stone are found to have, for every 
square inch of section, a cohesive power of 1811 lb. and 857 lb.; 
wherefore, suspended columns of these stones, of the altitude of 
1542 and 945 feet, or only the 1394th and 1789th part of their 
respective measure of elasticity, would be torn asunder by their 
own weight. 

31. Of the principal kinds of timber employed in building 
and carpentry, the annexed table will exhibit their respective 
Modulus of Elasticity, and the portion of it which limits their 
cohesion, or which lengthwise would tear them asunder. 



MODULUS OF ELASTICITY. 407 

Teak - - - - 6,040,000 feet 168th 

Oak - - - - 4,150,000 feet - - - 144th 

Sycamore - - - 3,860,000 feet 108th 

Beech - - - - 4,180,000 feet 107th 

Ash - - - - 4,617,000 feet 109th 

Elm - - - - 5,680,000 feet 146th 

Memelfir - - - 8,292,000 feet 205th 

Christiana deal - 8,118,000 feet 146th 

Larch - - - - 5,096,000 feet 121th 

The professor gives also this tabular view of their absolute 
cohesion, or the load which would rend a prism of an inch 
square ; and the altitude of the prism which would be severed 
by the action of its own weight. 

Teak - - - - 12,915 lb. 36,049 feet. 

Oak --- - 11,880 lb. 32,900 feet. 

Sycamore - - - 9,630 lb. 35,800 feet. 

Beech - - - - 12,225 lb. 38,940 feet. 

Ash - - - - 14,130 lb. 42,080 feet. 

Elm - - - - 9,720 lb. 39,050 feet. 

Memel fir - - 9,540 lb. 40,500 feet. 

Christiana deal - 12,346 lb. 55,500 feet. 

Larch - - - - 12,240 lb. 42,160 feet. 

32. The modulus of the elasticity of hempen fibres has not 
been ascertained, but may probably be reckoned about 5,000,000 
feet. Their cohesion is, for every square inch of transverse 
section, 6,400 lb. The best mode of estimating the strength 
of a rope of hemp is to multiply by 200, the square of its num- 
ber of inches in girth, and the product will express in pounds 
the practical strain it may be safely loaded with ; for cables 
multiply by 120, instead of 200. The ultimate strain is pro- 
bably double this ; as will appear from the account following 
of Du Hamel's experiments. If yarns of 180 yards long be 
worked up into a rope of only 120 yards, it will lose one- 
fourth of its strength, the exterior fibres alone resisting the 
greatest part of the strain. The register cordage of the late 
Captain Huddart exerts nearly the whole force of the strands, 
since they suffer a contraction of only the eighth part in the 
process of combining. 

33. For the utmost strength that a rope will bear before it 
breaks, a good estimate will be found by taking one-fifth 
of the square of the girth of the rope, to express the tons 
it will carry. This is about double the rule for practice which 
we have just given (art. 32) ; and is, even for an ulterior mea- 
sure, too great for tarred cordage, which is always weaker than 
white. 



408 STRENGTH OF CORDAGE. 

The following experiments were made by Mons. Du Hamel, 
at Rochfort, on cordage of 3-inch French circumference, made 
of the best Riga hemp, August 8th, 1741. 

White. Tarred. 

Broke with strain of 4500 pounds - - 3400 pounds. 
4000 - - - - 3300 
4800 - - - - 3258 

August 25th, 1743. 
4600 - - - - 3500 
5000 - - - - 3400 
5000 - - - - 3400 

September 23, 1746. 
3880 - - - - 3000 
4000 - - - - 2700 
4200 - - - - 2800 

A parcel of white and tarred cordage was taken out of a quan- 
tity which had been made Feb. 12th, 1746. 

It was laid up in the Magazine, and comparisons were made 
from time to time, as — 

White bore. 

1746 April 14th 2645 - - 

1747 May 18th 2762 - - 

1747 October 21st 2710 - - 

1748 June 19th 2575 - - 

1748 October 2d 2425 - - 

1749 Sept. 25th 2917 - - 

M. Du Hamel says, that it is decided by experience, that white 
cordage in continued service is one-third more durable than 
tarred ; secondly, it retains its force much longer while kept in 
store ; thirdly, it resists the ordinary injuries of the weather 
one-fourth longer. These observations deserve the attention of 
practical men. 

34. The metals differ more widely from each other in their 
elastic force and cohesive strength, than the several species of 
wood or vegetable fibres. Thus, the cohesion of fine steel is 
about 135,000 lb. for the square inch, while that of cast lead 
amounts only to about the hundred and thirtieth part, or 1800 lb. 

According to the accurate experiments of Mr. George Rennie 
in 1817, the cohesive power of a rod an inch square of different 
metals, in pounds avoirdupois, with the corresponding length in 
feet, is as follows : 

Cast steel ----- 134,2561b. -- 39,455 feet. 

Swedish malleable iron - 72,064 lb. -- 19,740 feet. 

English ditto - - 55,872 lb. -- 16,938 feet. 

Castiron 19,096 lb. - - 6,110 feet. 



Tarred. 


Difference. 


2312 


- - 333 


2155 


- - 607 


2050 


- - 660 


1752 


- - 823 


1837 


- - 588 


1865 


- - 1052 



MODULUS OP COHESION. 409 

Cast copper 19,072 lb. - - 5,003 feet. 

Yellow brass 17,958 lb. - - 5,180 feet. 

Cast tin 4,736 lb. - - 1,496 feet. 

Cast lead 1,824 lb. - - 348 feet. 

It thus appears, as Professor Leslie remarks, that a vertical 
rod of lead 348 feet long, would be rent asunder by its own 
weight. The best steel has nearly twice the strength of English 
soft iron, and this again is about three times stronger than cast 
iron. Copper and brass have almost the same cohesion as cast 
iron. This tenacity is sometimes considerably augmented by 
hammering or wire-drawing, that of copper being thus nearly 
doubled, and that of lead, according to Eytelwein, more than 
quadrupled. The consolidation is produced chiefly at the sur- 
face, and hence a slight notch with a file will materially weaken 
a hard metallic rod. English malleable iron has 7,550,000 
feet for its modulus of elasticity, or the weight of 24,920,000 
lb. on the square inch, while cast iron has 5,895,000 feet, and 
18,421,000 lb. Of other metals, the modulus of elasticity is 
probably smaller, but has not yet been well ascertained. 

35. The Longitudinal Compression which any column suf- 
fers, is at first equal to the dilation occasioned by an equal and 
opposite strain, being in both cases proportional to the modulus 
of elasticity. But while the incumbent weight is increased, the 
power of resistance likewise augments, as long as the column 
withstands flexure. After it begins to bend, a lateral disruption 
soon takes place. A slender vertical prism is hence capable of 
supporting less pressure than the tension which it can bear. 
Thus, a cubic inch of English oak was crushed only by the load 
of 3860 lb., but a bar of an inch square and 5 inches high gave 
way under the weight of 2572 lb. It would evidently have 
been still feebler if it had been longer. On the other hand, if 
the breadth of a column be considerable in proportion to its 
height, it will sustain a greater pressure than its cohesive power. 
Thus, though the cohesion of a rod of cast iron of a quarter of 
an inch square is only 300 lb., a cube of that dimension will re- 
quire 1440 lb. to crush it. 

In general, while the resisting mass preserves its erect form, 
the several sections are compressed and extended by additional 
weight, and their repellant particles are not only brought nearer, 
but multiplied. This repulsion is likewise increased by the la- 
teral action arising from the confined ring of detrusion. The 
primary resistance becomes hence greatly augmented in the pro- 
gress of loading the pillar. 

36. The most precise experiments on this subject, seem to be 

2N 



410 LONGITUDINAL COMPRESSION. 

those of Mr. Rennie. The weights required to crush cubes of 
the quarter of an inch of certain metals, are as follow : 

Iron cast vertically - -. - - 11,140 lb. 

Iron cast horizontally - - - - 10,110 lb. 

Cast copper ----- 7,318 lb. 

Cast tin ------ 9661b. 

Cast lead 483 lb. 

Cubes of an inch are crushed by the weights annexed : 

Elm 1,284 lb. 

White deal ----- i ? 928 lb. 

English oak 3,860 lb. 

Craigleith freestone - 8,688 lb. 

Cubes of an inch and a half, and consequently presenting a 
section of 2\ times greater than the former, might be expected 
to resist compression in that ratio. They are crushed, however, 
with loads considerably less. 

Red brick 1,817 lb. 

Yellow baked brick - 2,254 lb. 

Firebrick 3,864 lb. 

Craigleith stone, direction of the strata - 15,560 lb. 

Ditto, across the strata - 12,346 lb. 

White statuary marble - - - 13,632 lb. 

White-veined Italian marble - - 21,783 lb. 

Purbeck limestone - - - - 20,610 lb. 

Cornish granite - 14,302 lb. 

Peterhead granite - - - - 18,636 lb. 

Aberdeen blue granite - - - - 24,536 lb. 
These facts show the comparative firmness of different mate- 
rials, but it is to be regretted, that such results are not of much 
practical value, since they are confined to a very narrow scale, 
and applicable only to cubical blocks. While the breadth re- 
mains the same, the resistance appears to depend on some unas- 
certained ratio of the altitude of the column. 

Nay, as Professor Leslie observes, the absolute height it- 
self has probably a material influence on the effect. Thus, 
from some experiments made in France, it appears, that prisms 
of seasoned oak, two inches square, and two, four, or six feet 
high, would be crushed by the vertical pressures of 17,500 
lb., 10,500 lb., and 7,000 lb. ; but, if four inches square, and 
of the same altitudes, they would give way under loads of 
only 80,000 lb., 70,000 lb., and 50,000 lb. In the first set of 
trials, the mean cohesive power amounts to 130,000 lb., and 
in the second to 520,000 lb. The vertical support is there- 
fore greatly inferior to these limits. When the length of 
the pillar exceeds 36 times its breadth, the resistance to Ion- 



MODULUS OP COHESION. 411 

gitudinal compression appears to be diminished 18 times. — (Les- 
lie's Elements, and Duhamel in Mem. Paris. Jicad.) 

37. Mr. B. Bevan has favoured the author with a tabular 
view of his results with regard to the modulus of cohesion, or 
the length in feet of any prismatic substance required to break 
its cohesion, or tear it asunder. 

Bevan' *s Results. 

feet 
Tanned cow's skin - 10,250 



calfskin - 5,050 

horse skin - 7,000 

cordovan - 3,720 

sheep skin ... 5,600 

Untanned horse skin - 8,900 

Old harness of 30 years - - 5,000 
Hempen twine - - -'"-.- 75,000 

Catgut, some years old - - - 23,000 

Garden matting - 27,000 

Writing-paper, foolscap - - 8,000 

Brown wrapping-paper, thin - - 6,700 

Bent grass, (holcus) - 79,000 

Whalebone - 14,000 

Bricks, (Fenny Stratford) - - 970 

(Leighton) - - - 144 

Ice . . - 300 

Leicestershire slate - 7,300 

The following, also, exhibits Mr. Sevan's results as to the 
modulus of elasticity. 

feet. 

Platinum - 2,390,000 

Gold (pure) - 1,390,000 

Steel 9,300,000 

Bar-iron - 9,000,000 

Ditto 8,450,000 

Yellow pine - 9,150,000 

Ditto 11,840,000 

Finland deal - 6,000,000 

Mahogany - 7,500,000 

Rose wood - 3,600,000 

Oak, dry 5,100,000 

Fir bottom, 25 years old - - 7,400,000 

Petersburg deal - - - 6,000,000 

Lance wood - 5,100,000 

Willow 6,200,000 

Oak 4,350,000 



<*12 MODULUS OF COHESION, &C. 

feet. 
Satin wood - 2,290,000 

Lincolnshire bog oak - - 1,710,000 
Lignum vitse - 1,850,000 

Teak wood - - - 4,780,000 

Yew 2,220,000 

Whalebone - 1,000,000 

Cane 1,400,000 

Glass tube - 4,440,000 

Ice 6,000,000 

Limestone 

Dinton, Buck - - 2,400,000 

Ketton - - - 1,600,000 



Jetternoe - 635,000 

Ryegate 621,000 

Yorkshire paving - 1,320,000 

Cork 3,300 

Slate, Leicestershire - - 7,800,000 

Many other results are collected in Mr. TredgoWs Essay 
on the Strength of Cast Iron, a work which may be consulted 
with advantage on this and kindred topics. 

Indeed, we have not aimed at more in this section than a 
brief summary of the leading principles involved in the con- 
sideration of passive strength, and a corresponding exhibition 
of the best ascertained facts. The subject is one of great and 
growing interest to all concerned in the erection of extensive 
structures. Such may consult, in addition to the volume just 
mentioned, TredgoWs Carpentry, Barlow on the Strength 
of Timber, Gregory's Mechanics, Vol. I., the Lectures of Dr. 
T. Young and Professor Leslie, already quoted, and Girard, 
Traite Jlnalytique de la Resistance des Solids. 



SUPPLEMENTARY TABLES. 



TABLE I. — Useful Factors, extended to several places of 
Decimals, in which p represents the Circumference of a 
Circle, whose Diameter is 1. 

Then 

p = 3-1415926535897932384626433832795028841971 
6939937510582097494459230781640628620899 
8628034825342117067982148086513272306647 
0938446 + 
2p . . . 6-283185307179586476925286766559 
4p . . . 12-566370614359172953850573533118 
hp . . . 1-570796326794896619231321691639 
\p . . . 0-785398163397448309615660845819 
| p . . . 4-188790204786390984616857844372 
£p . . . 0-523598775598298873077107230546 
ip . . . 0-392699081698724154807830422909 
fVjO . . . 0-261799387799149436538553615273 
3^-oP • • • 0-008726646259971647884618453842 
1 

. 0-3183098861837906715377675267450 



. 0-636619772367581343075535053490 
. 1-273239544735162686151070106980 



P 

2 

P 
4 

P 

~ — ... 0-0795774715459476678844418816862 
4 p 

^/2 ... 1-41421356237309504880168S7242097 nearly 

s/h ... 0-707106781186547524400844362104 

p^/2 . . . 4-4428829381583662470158809900605 

p^/\ . . . 0-2214414690791831235079404950302 

-s/\ . . . 0-2250790790392765173887997751 
P 

-v/2 . . . 0-4501581580785530347775995955 
P 
s/p. . . 1-772453850905516027298167483341 

54 2 n 2 413 



414 SUPPLEMENTARY TABLES. 

\s/p ... 0*886226925452758013649083741670 

2^/p . . . 3-544907701811032054596334966682 

s/- • • • 1-253314137315500251207882642402 
2 

v/- . . . 0-797884560802865355879892119868 

s/~ . . . O-56418958354775628694807945156O 
V 

2s/- . . . 1-128379167095512573896158903120 
P 

\s/- ... 0-282094791773878143474039725780 
P 

f . . . 9-869604401089358618834490999876 



i> a 



I 

o 

1 

2/?* 

1 



6jo 



6^ 
360 



P. 

24 

6 



. 0-101321183642337771443879463209 
. 0-050660591821168885721939731604 
. 0-016886863940389628573979910534 



2s/— . . . 1-128379167095512573896158903120 
V 

-s/— . . . 0-094031597257959381158013241926 
6 p 

-s/- . . . 0-070523697943469535868509931445 
8 p 

1 



0-053051647697298445256294587790 



114-591559026164641753596309628200 



P 

\p . . . 2-094395102393195492308428922186 

. . 0-130899693899574718269276807636 



. 1-909859317102744029226605160470 
V 

V- . . . 1-2407009819. &c. 

P 

6p* . . . 59-217626406536151713006945999256 

V6p 2 . . . 3-89777707, &c. 

36p . . 113-097335529232556584655161798062 

&36p . . . 4-83597586204, &c. 

\s/p • . . 0-221556731363189503412270935418 



SUPPLEMENTARY TABLES. 



415 



1 
*P 

>/231 

v/282 



— • • • 0-159154943091895335768883763372 



0-805995977007, &c. 



15-1986841535706636 
16-7928556237466 + &c. 
n/10152. 100-75713374 

282 

v- 



■== 18-948708, &c. 



•785398, &c. 
v/277-274= 16-65154647474 
Log. of j»=0-49714987269413385435 



TABLE II. — A Table of Circles, from which, knowing the 
diameters, the areas, circumferences, and sides of equal 
squares are found. 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


1-00 


0-7853981 


3-14159265 


0-88622692 


•25 


1-22718463 


3-92699081 


1-10778365 


•5 


1-76714586 


4-71238898 


1-32934038 


•75 


2-40528187 


5-49778714 


1-55089711 


2. 


3-14159265 


6-28318530 


1-77245384 


•25 


3-97607820 


7-06858347 


1-99401058 


•5 


4-90873852 


7-85398163 


2-21556731 


•75 


5-93957361 


8-63937979 


2-43712404 


3- 


7-06858347 


9-42477796 


2-65868077 


•25 


8-29576810 


10-21017612 


2-88023750 


•5 


9-62112750 


10-99557428 


3-10179423 


•75 


11-04466167 


11-78097245 


3-32335096 


4- 


12-56637061 


12-56637061 


3-54490769 


•25 


14-18625432 


13-35176877 


3-76646442 


•5 


15-90431280 


14-13716694 


3-98802116 


•75 


17-72054606 


14-92256510 


4-20957789 


5- 


19-63495408 


15-70796326 


4-43113462 


•25 


21-64753687 


16-49336143 


4-65269135 


•5 


23-75829444 


17-27875959 


4-87424808 


•75 


25-96722677 


18-06415775 


5-09580482 


6- 


28-27433388 


18-84955592 


5-31736155 


•25 


30-67961575 


19-63495408 


5-53891828 


•5 


33-18307240 


20-42035224 


5-76047501 


•75 


35-78470382 


21-20575041 


5-98203174 


7- 


38-48451001 


21-99114857 


6-20358847 


•25 


41-28249096 


22-77654673 


6-42514520 


•5 


44-17864669 


23-56194490 


6-64670193 


•75 


47-17297718 


24-34734306 


6-86825866 


8- 


50-26548245 


25-13274122 


7-08981539 


•25 


53-45616249 


25-91813939 


7-31137213 


•5 


56-74501730 


26-70353755 


7-53292886 


•75 


60.13204688 


27-48893571 


7-75448559 



416 



SUPPLEMENTARY TABLES. 



Given. 
Diam. 


Required. 


Area. 


| Circumference. 


side of eq. sq. 


9- 


63-61725123 


28-27433388 


7-97604232 


•25 


67-20063035 


29-05973204 


8-19759905 


•5 


70-88218424 


29-84513020 


8-41915578 


•75 


74-66191290 


30-63052837 


8-64071251 


10- 


78-53981633 


31-41592653 


8-86226925 


•25 


82-51589454 


32-20132469 


9-08382598 


•5 


86-59014751 


32-98672286 


9-30538270 


•75 


90-76257525 


33-77212102 


9-52693944 


11- 


95-03317777 


34-55751918 


9-74849617 


•25 


99-40195505 


35-34291735 


9-97005290 


•5 


103-86890711 


36-12831551 


10-19160964 


•75 


108-43403393 


36-91371367 


10-41316637 


12- 


113-09733553 


37-69911184 


10-63472310 


•25 


117-85881189 


38-48451000 


10-85627983 


•5 


122-71846303 


39-26990816 


11-07783656 


•75 


127-67628893 


40-05530633 


11-29939329 


13- 


132-73228961 


40-84070449 


11-52095002 


•25 


137-88646506 


41-62610265 


11-74250675 


•5 


143-13881527 


42-41150082 


11-96406348 


•75 


148-48934026 


43-19689898 


12-18562021 


14- 


153-93804002 


43-98229714 


12-40717695 


•25 


159-48491455 


44-76769531 


12-62873368 


•5 


165-12996385 


45-55309347 


12-85029041 


•75 


170-87318792 


46-33849163 


13-07184714 


15* 


176-71458676 


47-12388980 


13-29340388 


•25 


182-65416028 


47-90928796 


13-51496061 


•5 


188-69190875 


48-69468613 


13-73651734 


•75 


194-82783190 


49-48008429 


13-95807407 


16- 


201-06192982 


50-26548245 


14-17963080 


•25 


207-S9420252 


51-05088062 


14-40118753 


•5 


213-82464998 


51-83627878 


14-62274426 


•75 


220-35327221 


52-62167694 


14-84430099 


17- 


226-98006922 


53-40707511 


15-06585772 


•25 


233-70504099 


54-19247327 


15-28741446 


•5 


240-52818753 


54-97787143 


15-50897119 


•75 


247-44950885 


55-76326960 


15-73052792 


18- 


264-46900494 


56-54866776 


15-95208465 


•25 


266-58667578 


57-33406592 


16-17364138 


•5 


268*80252140 


58-11946409 


16-39519811 


•75 


276-11654180 


58-90486225 


16-61675484 


19- 


283-52873699 


59-69026041 


16-83831157 


•25 


291-03910692 


60-47565858 


17-05986830 


•5 


298-64765163 


61-26105674 


17-28142503 


•75 


306-35437111 


62-04645490 


17-50298177 


20- 


314-15926535 


62-83185307 


17-72453850 


•25 


322-06233437 


63-61725123 


17-94609524 


•5 


330-06357816 


64-40264939 


18-16765197 


•75 


338-16299672 


65-18804756 


18-38920870 


21- 


346-36059005 


65-97344572 


18-61076543 


•25 


354-65635814 


66-75884388 


18-83232216 


•5 


363-05030101 


67-54424205 


19-05387889 


•75 


371-54241865 


68-32964021 


19-27543562 


22- 


380-13271108 


69-11503837 


19-49699235 


•25 


388-82117826 


69-90043654 


19-71854908 


•5 


397-60782021 


70-68583470 


19-94010581 


•75 


406-49263694 


71-47123286 


20-16166255 



SUPPLEMENTARY TABLES. 



417 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


23- 


415-47562843 


72-25663103 


20-38321928 


•25 


424-55679467 


73-04202919 


20-60477601 


•5 


433-73613573 


73-82742735 


20-82633274 


•75 


443-01365154 


74-61282552 


21-04788945 


24- 


452-38934212 


75-39822368 


21-26944618 


•25 


461-86320745 


76-18362184 


21-49100291 


•5 


471-43524757 


76-96902001 


21-71255964 


•75 


481-10546239 


77-75441817 


21-93411637 


25* 


490-87385212 


78-53981634 


22-15567313 


•25 


500-74041655 


79-32521450 


22-37722986 


•5 


510-70515575 


80-11061266 


22-59878659 


•75 


520-76806971 


80-89601083 


22-82034332 


26' 


530-92915845 


81-68140899 


23-04190006 


•25 


541-18842196 


82-46680715 


23-26345679 


•5 


551-54586024 


83-25220532 


23-48501352 


•75 


562-00147328 


84-03760348 


23-70657025 


27' 


572-55526112 


84-82300164 


23-92812698 


•25 


583-20722369 


85-60839981 


24-14968371 


•5 


593-95736105 


86-39379797 


24-37124044 


•75 


604-80567318 


87-17919613 


24-59279717 


28- 


615-75216010 


87-96459430 


24-81435390 


•25 


626-79682177 


88-74999246 


25-03591063 


•5 


637-93965822 


89-53539062 


25-25746737 


•75 


649-18066943 


90-32078879 


25-47902410 


29' 


660-51985541 


91-10618695 


25-70058083 


•25 


671-95721616 


91-89158511 


25-92213756 


•5 


683-49275169 


92-67698328 


26-14369429 


•75 


695-12646198 


93-46238144 


26-36525102 


30* 


706-85834706 


94-24777960 


26-58680776 


•25 


718-68840688 


95-03317777 


26-80836449 


•5 


730-61664148 


95-81857593 


27-02992122 


•75 


742-64305085 


96-60397409 


27-25147794 


31- 


754-76763502 


97-38937226 


27-47303468 


•25 


766-99039394 


98-17477042 


27-69459141 


•5 


779-31132762 


98-96016858 


27-91614814 


•75 


791-73043607 


99-74556675 


28-13770488 


32- 


804-24771932 


100-53096491 


28-35926161 


•25 


816-86317729 


101-31636307 


28-58081834 


•5 


829-57681005 


102-10176124 


28-80237507 


•75 


842-38861759 


102-88715940 


29-02393180 


33- 


855-29859994 


103-67255756 


29-24548853 


•25 


868-30675696 


104-45795573 


29-46704526 


•5 


881-41308881 


105-24335389 


29-68860199 


•75 


894-61759542 


106-02875205 


29-91015872 


34* 


907-92027688 


106-81415022 


30-13171545 


•25 


921-32113305 


107-59954838 


30-35327219 


•5 


934-82016398 


108-38494654 


30-57482892 


•75 


948-41736968 


109-17034471 


30-79638565 


35- 


962-11275016 


109-95574287 


31-01794239 


•25 


975-90630540 


110-74114103 


31-23949912 


•5 


989-79803541 


111-52653920 


31-46105585 


•75 


1003-78794019 


112-31193736 


31-68261258 


36- 


1017-87601975 


113-09733552 


31-90416931 


•25 


1032-06227407 


113-88273369 


32-12572604 


•5 


1046-34670316 


114-66813185 


32-34728277 


•75 


1060-72930703 


115-45353001 


32-56883950 



418 



SUPPLEMENTARY TABLES. 



Given. 


Required. 


Diam. 


Area. 


1 Circumference. 


side of eq. sq. 


37- 


1075-21008569 


116-23892818 


32-79039623 


•25 


1089-78903909 


117-02432634 


33-01195296 


•5 


1104-46616727 


117-80972450 


33-23350970 


•75 


1119-24147022 


118-59572267 


33-45506643 


38- 


1134-11494794 


119-38052083 


33-67662316 


•25 


1149-08660043 


120-16591899 


33-89817989 


•5 


1164-15642768 


120-95131716 


34-11973662 


-75 


1179-32442971 


121-73671532 


34-34129335 


39* 


1194-59060653 


122-52211348 


34-56285008 


•25 


1209-95495809 


123-30751165 


34-78440681 


•5 


1225-41748449 


124-09290981 


35-00596354 


•75 


1240-97818552 


124-87830797 


35-22752027 


40' 


1256-63706144 


125-66370614 


35-44907701 


•25 


1272-39411208 


126-44910430 


35.67063374 


•5 


1288-24933751 


127-23450246 


35-89219048 


•75 


1304-20273770 


128-01990063 


36-11374721 


41- 


1320-25431266 


128-80529879 


36-33530394 


•25 


1336-40406240 


129-59069695 


36-55686067 


•5 


1352-65198690 


130-37609512 


36-77841740 


•75 


1368-99808617 


131-16149328 


36-99997413 


42- 


1385-44236022 


131-94689144 


37-22153086 


•25 


1401-98480903 


132-73228961 


37-44308759 


•5 


1418-62543261 


133-51768777 


37-66464432 


•75 


1435-36423096 


134-30308593 


37-88620105 


43- 


1452-20120412 


135-08348410 


38-10775779 


•25 


1469-13635202 


135-87388226 


38-32931452 


•5 


1486-16967468 


136-65928042 


38-55087125 


•75 


1503-30117212 


137-44467859 


38-77242798 


44- 


1520-53084433 


138-23007675 


38-99398471 


•25 


1537-85869131 


139-01547491 


39-21554144 


•5 


1556-28471306 


139-80087308 


39-43709817 


•75 


1572-80890957 


140-58627124 


39-65865490 


45- 


1590-43128088 


141-37166940 


39-88021164 


•25 


1608-15182692 


142-15706757 


40-10176837 


•5 


1625-97054775 


142-94246573 


40-32332510 


•75 


1643-88744335 


143.72786390 


40-54488183 


46- 


1661-90251374 


144-51326206 


40-76643856 


•25 


1680-01575889 


145-29866022 


40-98799530 


•5 


1698-22717880 


146-08405839 


41.20955203 


•75 


1716-53677348 


146-86945655 


41-43110876 


47- 


1734-94454294 


147-65485471 


41-65266549 


•25 


1753-45048716 


148-44025288 


41-87422222 


•5 


1772-05460615 


149-22565104 


42-09577891 


•75 


1790-75689992 


150-01104920 


42-31733568 


48- 


1809-55736847 


150-79644797 


42-53889241 


•25 


1828-45601175 


151-58184553 


42-76044914 


•5 


1847-45282982 


152-36724369 


42.98200587 


•75 


1866-54782267 


153-15264186 


43-20356261 


49- 


1885-74099031 


153-93804002 


43-42511934 


•25 


1905-83233270 


154-72343818 


43-64667607 


•5 


1924-42184986 


155-50883635 


43-86823280 


•75 


1943-90954179 


156-29423451 


44-08978953 


50- 


1963-49540848 


157-07963268 


44-31134627 


•25 


1983-17944995 


157-96503084 


44-53290300 


5 


2002-96166619 


158-65042900 


44-75445973 


•75 1 


2022-84205720 


159-43582717 


44-97601646 



SUPPLEMENTARY TABLES. 



419 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


51- 


2042-82062300 


160-22122533 


45-19757319 


•25 


2062-89736352 


161-00662349 


45*41912992 


•5 


2083-07227884 


161-79202166 


45-64068665 


•75 


2103-34536893 


162-57741982 


45-86224338 


52- 


2123-71663382 


163-36281798 


46-08380012 


•25 


2144-18607346 


164-14821615 


46-30535685 


•5 


2164-75368786 


164-93361431 


46-52691358 


•75 


2185-41947703 


165-71901247 


46-74847031 


53' 


2206-18344098 


166-50441064 


46-97002704 


•25 


2227.04557969 


167-28980880 


47-19158377 


•5 


2248-00589318 


168-07520696 


47-41314050 


•75 


2269-06438143 


168-86060513 


47-63469723 


54- 


2290-22104447 


169-64600329 


47-85625386 


•25 


2311-47588225 


170-43140145 


48-07781069 


•5 


2332-82889481 


171-21679962 


48-29936743 


•75 


2354-28008215 


172-00219778 


48-52092416 


55- 


2375-82944427 


172-78759594 


48-74248089 


•25 


2397-47698115 


173-57299411 


48-96403763 


•5 


2419-22269280 


174-35839227 


49-18559436 


•75 


2441-06657922 


175-14379043 


49-40715109 


56- 


2463-00864041 


175-92918860 


49-62870782 


•25 


2485-04887637 


176-71458676 


49-85026455 


•5 


2507-18728710 


177-49998492 


50-07182128 


•75 


2520-42387260 


178-28538309 


50-29337801 


57* 


2551-75863288 


179-07078125 


50-51493474 


•25 


2574-19156790 


179-85617941 


50-73649147 


•5 


2596-72257781 


180-64157758 


50-95804820 


•75 


2619-35196239 


181-42697574 


51-17960494 


58- 


2642-07942166 


182-21237390 


51-40116167 


•25 


2664-90505579 


182-99777207 


51-62271840 


•5 


2687-82886464 


183-78317023 


51-84427513 


•75 


2710-85084834 


184-56856839 


52-06583186 


59- 


2733-97100678 


185-35396656 


52-28738859 


•25 


2757-18933998 


186-13936472 


52-50894532 


•5 


2780-50584795 


186-92476288 


52-73050205 


•75 


2803-92058070 


187-71016105 


52-95205878 


60- 


2827-43338828 


188-49555921 


53-17364552 


•25 


2851-04442049 


189-28095737 


53-39517225 


•5 


2874-75362754 


190-06635554 


53-61672898 


•75 


2898-56100937 


190-85175370 


53-83828572 


61- 


2922-46656592 


191-63715186 


54-05984245 


•25 


2946-47029734 


192-42255003 


54-28139918 


•5 


2970-57220350 


193-20794819 


54-50295591 


•75 


2994-77228444 


193-99334635 


54-72451264 


62* 


3019-07054010 


194-77874452 


54-94606937 


•25 


3043-46697053 


195-56414268 


55-16762610 


•5 


3067-96157576 


196-34954084 


55-38918283 


•75 


3092-55435572 


197-13493901 


55-61073956 


63- 


3117-24531051 


197-92033717 


55-83229629 


•25 


3142-03444002 


198-70573533 


56-05385303 


•5 


3166-92174434 


199-49113350 


56-27540976 


•75 


3191-90722341 


200-27653166 


56-49696649 


64- 


3216-99087728 


201-06192982 


56-71852322 


•25 


3242-17270581 


201-84732799 


56-94007995 


•5 


3267-45270920 


202-63272615 


57-16163668 


•75 


3292-83088742 


203-41812431 


57-38319341 



420 



SUPPLEMENTARY TABLES. 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


65- 


3318-30724035 


204-20352248 


57.60475015 


•25 


3343-88176802 


204-98892064 


57-82630688 


•5 


3369-55447036 


205-77431880 


58-04786361 


•75 


3395-32534774 


206-55971697 


58-26942034 


66- 


3421-19439976 


207-34511513 


58-49097707 


•25 


3447-16182652 


208-13051329 


58-71253380 


•5 


3473-22702806 


208-91591146 


58-93409054 


•75 


3499-39060438 


209-70130962 


59-15564727 


67* 


3525-65235549 


210-48670778 


59-37720400 


•25 


3552-01228735 


211-27210595 


59.59876073 


•5 


3578-47038197 


212-05750411 


59-82031746 


•75 


3605-02665737 


212-84290227 


60-04187419 


68' 


3631-68110754 


213-62830044 


60-26343092 


•25 


3658-43373248 


214-41369860 


60-48498765 


•5 


3685-28453219 


215-19909676 


60-70654438 


•75 


3712-23350667 


215-98449493 


60-92810111 


69- 


3739-28065594 


216-76989309 


61-14965785 


•25 


3766-42597994 


217-55529125 


61-37121458 


•5 


3793-66947873 


218-34068942 


61-59277131 


•75 


3821-01115229 


219-12608758 


61-81432804 


70- 


3848-45100064 


219-91148574 


62-03588478 


•25 


3875-98902375 


220-69688391 


62-25744151 


•5 


3903-62522162 


221-48228207 


62-47899824 


•75 


3931-35959426 


222-26768023 


62-70055497 


71- 


3959-19214168 


223-05307840 


62-92211170 


•25 


3987-12286386 


223-83847656 


63-14366843 


•5 


4015-15176082 


224-62387472 


63-36522516 


•75 


4043-27883254 


225-40927289 


63-58678189 


72- 


4071-40507905 


226-19467105 


63-80833862 


•25 


4099-82750030 


226-98006921 


64-02989536 


•5 


4128-24909633 


227-76546738 


64-25145209 


•75 


4156-76886714 


228-55086554 


64-47300882 


73- 


4185-38681274 


229-33626370 


64-69456555 


•25 


4214-10293309 


230-12166187 


64-91612228 


•5 


4242-91722821 


230-90706003 


65-13767901 


•75 


4271-02969810 


231-69245819 


6535923574 


74- 


4300-84034275 


232-47785636 


65-58079247 


•25 


4329-94916219 


233-26325452 


65-80234920 


•5 


4359-15615638 


234-04865268 


66-02390593 


•75 


4388-46132535 


234-83405085 


66-24546267 


75- 


4417-86466910 


235-61944901 


66-46701940 


•25 


4447-36618760 


236-40484717 


66-68857613 


•5 


4476-96588088 


237-19024534 


66-91043287 


•75 


4506-66374893 


237-97564350 


67-13168960 


76- 


4536-45979178 


238-76104166 


67-35324633 


•25 


4566-35400937 


239-54643983 


67-57480306 


•5 


4596-34640174 


240-33183799 


67-79635979 


•75 


4626-43696897 


241-11723615 


68-01791652 


77- 


4656-62571078 


241-90263432 


68-23947325 


•25 


4686-91262745 


242-68803248 


68-46102998 


•5 


4717-29771890 


243-47343064 


68-68258671 


•75 


4747-78098511 


244-25882881 


68-90414344 


78- 


4778-36242610 


245-04422697 


69-12570018 


•25 


4809-04204185 


245-82962513 


69-34725691 


•5 


4839-81983238 


246-61502330 


69-56881364 


•75 


4870-79579767 


247-40042146 


69-79037037 



SUPPLEMENTARY TABLES. 



421 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


79- 


4901-66993776 


248-18581962 


70-01192710 


•25 


4932-74225260 


248-97121779 


70-23348383 


•5 


4963-91274221 


249-75661595 


70-45504056 


•75 


4995-18140659 


250-34201411 


70-67659729 


80- 


5026-54824574 


251-32741228 


70-89815403 


•25 


5058-01325966 


252-11281044 


71-11971076 


•5 


5089-57644835 


252-89820860 


71-34126749 


•75 


5121-23781181 


253-68360677 


71-56282422 


81* 


5152-99735004 


254-46900493 


71-78438096 


•25 


5184-85506304 


255-25440309 


72-00593769 


•5 


5216-81095081 


256-03980126 


72-22749442 


•75 


5248-86501335 


256-82579942 


72-44905115 


82- 


5281-01725068 


257-61059758 


72-67060788 


•25 


5313-26766276 


258-39599575 


72-89216461 


•5 


5345-61624962 


259-18139391 


73-11372134 


•75 


5378-06301124 


259-96679207 


73-33527807 


83- 


5410-60794764 


260-75219024 


73-55683480 


•25 


5443-25105880 


261-53758840 


73-77839153 


•5 


5475-99234474 


262-32298656 


73-99994827 


•75 


5508-83180544 


263-10838473 


74-22150500 


84- 


5541-76944092 


263-89378269 


74-44306173 


•25 


5574-80525116 


264-67918105 


74-66461846 


•5 


5607-93923618 


265-46457922 


74-88617519 


•75 


5641-17139596 


266-24997738 


75-10773192 


85- 


5674-50173054 


267-03537554 


75-32928866 


•25 


5707-93023987 


267-82077371 


75-55084539 


•5 


5741-45692397 


268-60617187 


75-77240212 


•75 


5775-08178284 


269-39157003 


75-99395885 


86- 


5808-80481648 


270-17696820 


76-21551558 


•25 


5842-62602489 


270-96236636 


76-43707232 


•5 


5876-54540807 


271-74776452 


76-65862904 


•75 


5910-56296602 


272-53316269 


76-88018578 


87- • 


5944-67869876 


273-31856085 


77-10174251 


•25 


5978-89260623 


274-10395901 


77-32329924 


•5 


6013-20468849 


274-88935718 


77-54485597 


•75 


6047-61494552 


275-67475534 


77-76641270 


88- 


6082-12337734 


276-46015350 


77-98796943 


•25 


6116-72998392 


277-24555167 


78-20952616 


•5 


6151-43476526 


278-03094983 


78-43103289 


•75 


6186-23772138 


278-81634799 


78-65263962 


89- 


6221-13885226 


279-60174616 


78-87419635 


•25 


6256-13815792 


280-38714432 


79-09575309 


•5 


6291-23563834 


281-17254248 


79-31730982 


•75 


6326-43129354 


281-95794065 


79-53886655 


90' 


6361-72512352 


282-74333881 


79-76042329 


•25 


6397-11712824 


283-52873697 


79-98198002 


•5 


6432-60730774 


284-31413514 


80-20353673 


•75 


6468-19566202 


285-09953330 


80-42509348 


91- 


6503-88219109 


285-88493146 


80-64669021 


•25 


6539-66689491 


286-67032963 


80-86820694 


•5 


6575-54977350 


287-45572779 


81-08976367 


75 


6611-53082686 


288-24112595 


81-31132040 


92- 


6647-61005499 


289-02652412 


81-53287713 


•25 


6683-78745789 


289-81192228 


81-75443366 


•5 


6720-06303556 


290-59732044 


81-97599060 


•75 


6756-43678800 


291-38271861 


82-19754733 


55 


* 


I 





422 



SUPPLEMENTARY TABLES. 



Given. 


Required. 


Diam. 


Area. 


Circumference. 


side of eq. sq. 


93- 


6792-90871521 


292-16811677 


82-41910406 


•25 


6S29-47SS1719 


292-95351493 


82-64066079 


•5 


6866-14709394 


293-73891310 


82-86221752 


•75 


6902-91354546 


294-52431126 


83-08377425 


94- 


6939-77817177 


295-30970942 


83-30533096 


•25 


6976-74097284 


296-09510759 


83-52688771 


•5 


7013-80194367 


296-88050579 


83-74844444 


•75 


7050-96109923 


297-66590391 


83-97000117 


95- 


7088-21842465 


298-45130208 


84-19155791 


•25 


7125-57992480 


299-23670024 


84-41311464 


•5 


7163-02759971 


300-02209840 


84-63467136 


•75 


7200-57944940 


300-80749657 


84-85622811 


96- 


7238-22947387 


301-59289473 


85-07778484 


•25 


7275-97767308 


302-37829269 


85-29934157 


•5 


7313-82404707 


303-16369106 


85-52089830 


•75 


7351-76859584 


303-94908922 


85-74245503 


97- 


7389-81131940 


304-73448738 


85-96401176 


•25 


7427-95221771 


305-51988555 


86-18556849 


•5 


7466-19129079 


306-30528371 


86-40712522 


•75 


7504-52853864 


307-09068187 


86-62868195 


98. 


7542-96396126 


307-87608004 


86-85023869 


•25 


7581-49755865 


308-66147820 


87-07179542 


•5 


7620-12933081 


309.44687635 


87-29335215 


•75 


7658-85927774 


310-23227453 


87-51490888 


99- 


7697-68739946 


311-01767269 


87-73646568 


•25 


7736-61369591 


311-80307085 


87-95802234 


•5 


7775-63816715 


312-58846902 


88-17957907 


•75 


7814-76081316 


313-37386718 


88-40113580 


100- 


7853-98163397 


314-15926535 


88-62269254 


•25 


7893-30062952 


314-94466352 


88-84424927 


•5 


7932-71779985 


315-73006168 


89-06580600 


•75 


7972-23314494 


316-51545984 


89-28736273 



The preceding tables were computed with great care by the 
author's esteemed friend, the late H. Goodwin, Esq. of Black- 
heath, a gentleman whose indefatigable perseverance and re- 
markable accuracy in reference to numerical computations 
cannot be too highly characterized. They are inserted here to 
supersede the necessity of consulting some erroneous tables of 
the areas, &c. of circles recently put into circulation. 

Supposing the unit to be an inch, for example, the last table 
exhibits the area, circumference, and side of equal square, cor- 
responding to diameters varying by a quarter of an inch, from 
1 inch to 100. 

But the same table may also be made to serve for many inter- 
mediate diameters, as well as for diameters beyond its apparent 
reach, by simply recollecting that the circumferences and sides 
of equal squares are as the diameters, while the areas are as 
the squares of the diameters. 



USE OF TABLE OF CIRCLES. 423 



Thus, divide any 

2 

A 3 
circumi. ) . 

or side of > by - 

equal sq. ) 



10 &c. 



the quotient 

will give 

the 



mi 



circumf. 

or 

side of J i I .jEfo 
equal { p § 

square I } \ v hs 

L to LtVJ 



XI 



Or multiply by 2, 3, 4, 5, &c, the product will give the cir- 
cumference or side of equal square, to 2, 3, 4, 5, &c. times the 
assumed diameter. 

For areas, take i, -|, T \, ^V? & c - f° r i? h 4> h & c - tne as " 
sumed diameter, and, contrarily 4 times, 9 times, 16 times, 25 
times, 100 times, &c. an area, fcr the one which agrees to twice, 
3 times, 4 times, 5 times, 10 times, &c. the diameter to which 
the assumed area corresponds. 

Example. Find the area of 
8-125. 

Area to diameter 16-24 is 207*39420252 
Divide this by 2 2 = 4 



The quotient is the area required 51*84855063 



Ex. 2. Find the area of a circle whose diameter is 8*1, 

Area to diameter 81, is 5152*99735004 
Divide this by 10 3 = 100 



The quotient is the area required 51*5299735004 

Ex. 3. The exterior and the interior diameters of a circular 
ring, are 8*75 and 8*1. Required its area. 

Area to diameter 8*75, from table 60*13204688 



1, from above 51*52997350 



Their diff. is the area of the ring 8*60207338 

Areas of circular rings thus found, being multiplied into their 
thickness, will give their capacity, as in the rim of a fly-wheel, 
&c. ; whence, knowing the weight of a cubic inch, or foot, the 
weight of the whole becomes known : but the method already 
given at p. 300, will usually be found preferable, unless great 
accuracy be required ; in which case recourse may be had to 
this table. 



424 



SUPPLEMENTARY TABLES. 



Table III. — Relations of the Arc, Abscissa, Ordinate, and 
Subnornal, in the Catenary ; useful in the construction 
of Catenarian equilibrated arches for bridges or 'powder 
magazines. 



Arc 


Abscissa 


Ordinate 


Subnormal 


Arc 


Abscissa 


Ordinate 


Subnormal 


A P 


A G 


G P 


G K 


A P 


A G 


G P 


G K 


1 


l'OO 


0-02 


25-00 


40 


31-22 


22-17 


19-51 


2 


2-00 


0-08 


24-97 


41 


31-74 


23-02 


19-36 


3 


2-99 


0-18 


24-94 


42 


32-26 


23-88 


19-20 


4 


3-99 


0*32 


24-91 


43 


32-77 


24-74 


19-05 


5 


4-97 


0-50 


24-86 


44 


33-27 


25-61 


18-90 


6 


5-95 


0-71 


24-79 


45 


33-76 


26-48 


18-75 


7 


6-92 


0-96 


24-71 


46 


34-24 


27-36 


18-61 


8 


7-87 


1-25 


24-60 


47 


34-71 


28-24 


18-46 


9 


8-82 


T57 


24-50 


48 


35-18 


29-12 


18-32 


10 


9-75 


1-93 


24-37 


49 


35-64 


30-01 


18-18 


11 


10-67 


2*31 


24*25 


50 


36-09 


30-90 


18-05 


12 


11-58 


2-73 


24-12 


51 


36-53 


31-80 


17-91 


13 


12-47 


3*18 


23-9S 


52 


36-97 


32-70 


17-77 


14 


13-36 


3-65 


23-86 


53 


37-40 


33-60 


17-64 


15 


14-23 


4-16 


23-72 


54 


37-82 


34-51 


17-51 


16 


15-07 


4-68 


23-55 


55 


38-24 


35-42 


17-38 


17 


15-90 


5-23 


23-38 


56 


38-65 


36-33 


17-25 


18 


16-72 


5-81 


23-22 


57 


39-06 


37*24 


17-13 


19 


17-52 


6-40 


23-05 


58 


39-46 


38-16 


17-01 


20 


18-31 


7-01 


22-89 


59 


39-85 


39-08 


16-89 


21 


19-08 


7-65 


22-72 


60 


40-24 


40-00 


16-77 


22 


19-84 


8-31 


22-55 


61 


40-62 


40-93 


16-65 


23 


20-59 


8-98 


22-38 


62 


40-99 


41-85 


16-53 


24 


21-32 


9-66 


2221 


63 


41-36 


42-78 


16-42 


25 


22-03 


10-36 


22-03 


64 


41-73 


43-71 


16-30 


26 


22-73 


11-08 


21-86 


65 


42-09 


44-65 


16-19 


21 


23-42 


11-80 


21-68 


66 


42-45 


45-58 


16-08 


28 


24-09 


12-54 


21-51 


67 


42-80 


46-51 


15-97 


29 


24-76 


13-29 


21-34 


68 


43-15 


47-45 


15-86 


30 


25-40 


14-05 


21-17 


69 


43-49 


48-39 


15-76 


31 


26-04 


14-83 


21-00 


70 


43-83 


49-33 


15-65 


32 


26-66 


15-61 


20-83 


71 


44-17 


50-28 


15-55 


33 


27-27 


16-41 


20-66 


72 


44-50 


51-22 


15-45 


34 


27-87 


17-21 


20-49 


73 


44*82 


52-17 


15-35 


35 


28-45 


18-02 


20-33 


74 


45-14 


53-11 


15-25 


36 


29-03 


18-83 


20-16 


75 


45*46 


54-06 


15-16 


37 


29-59 


19-66 


20-00 


76 


45-77 


55-01 


15-06 


38 


30-14 


20-49 


19-83 


77 


46-08 


55-96 


14-96 


39 


30*68 


21-33 


19-67 


78 


46-39 1 


56-91 


14-87 



SUPPLEMENTARY TABLES. 



425 



Table III. — Relations of the 


Catenary, continued. 


Arc 


Abscissa 


Ordinate 


Subnormal 


Arc 


Abscissa 


Ordinate 


Subnormal 


A P 


A G 


G p 


G K 


A P 


A G 


G P 


G K 


79 


46-69 


57-86 


14-77 


115 


55-77 


92-68 


12-13 


80 


46-99 


58-81 


14-68 


116 


55-98 


93-66 


12-07 


81 


47-29 


59-77 


14-59 


117 


56-19 


94-64 


1201 


82 


47-58 


60-72 


14-50 


118 


56-39 


95-62 


11-95 


83 


47-87 


61-68 


14-42 


119 


56-60 


96-59 


11-90 


84 


48*16 


62-64 


14-33 


120 


56-80 


97-57 


11-84 


85 


48-44 


63-60 


14-25 


121 


57-00 


98-55 


11-78 


86 


48-73 


64-56 


14-16 


122 


57-21 


99-53 


11-73 


87 


49-00 


65-52 


14-08 


123 


57-41 


100-51 


11-67 


88 


49-28 


66-48 


14-00 


124 


57-61 


101-49 


11-62 


89 


49-55 


67-44 


13-92 


125 


57-81 


102-47 


11-57 


90 


49-82 


68-41 


13-84 


126 


58-00 


103-45 


11-51 


91 


5009 


69-37 


13-76 


127 


58-20 


104-42 


11-46 


92 


50-35 


70-34 


13-68 


128 


58-39 


105-42 


11-41 


93 


50-61 


71-30 


13-60 


129 


58-58 


106-40 


11-36 


94 


50*87 


72-27 


13-53 


130 


58-77 


107-38 


11-30 


95 


51*13 


73-24 


13-45 


131 


58-96 


108-35 


11-25 


96 


51*38 


74-20 


13-38 


132 


59-15 


109-35 


11-20 


97 


51*63 


75-17 


13-31 


133 


59-34 


110-33 


11-15 


98 


51-88 


76-14 


13-23 


134 


59-52 


111-31 


11-10 


99 


52-13 


77-11 


13-16 


135 


59-71 


112-28 


11-06 


100 


52-37 


78-08 


13-09 


136 


59-89 


113-28 


11-01 


101 


52-61 


79-05 


13-02 


137 


60-07 


114-26 


10-96 


102 


52S5 


80-02 


12-95 


138 


60-25 


115-25 


10-91 


103 


53*09 


81-00 


12-89 


139 


60-43 


116-23 


10-87 


104 


53-32 


81-97 


1282 


140 


60-60 


117-22 


10-82 


105 


53-55 


82-94 


12-75 


141 


60-77 


118*20 


10-77 


106 


53-78 


83-91 


12-69 


142 


60-95 


119-18 


10*73 


107 


54-01 


84-89 


12-62 


143 


61-12 


120*15 


1068 


108 


54*24 


85-86 


12-56 


144 


61-29 


121-15 


10-64 


109 


54-47 


86-83 


12*50 


145 


61-46 


122-14 


10-59 


110 


54-69 


87-81 


12-43 


146 


61*63 


123-12 


10-55 


111 


54*91 


88-78 


12-37 


147 


61-79 


12411 


10*50 


112 


55-13 


89-76 


12-31 


148 


61*96 


125-10 


10*46 


113 


55-35 


90-73 


12*25 


149 


62-12 


126-09 


10-42 


114 


55-56 91-71 | 


12-19 


150 


62-29 


127*08 


10-38 



At p. 179 of this volume there is given a table of relations 
of catenarian curves, so arranged as to be useful in the erection 
of suspension bridges, and in determining the weight and 
tension of the several parts. The table now presented in this 
supplement agrees with that in principle, but presents a totally 

2o2 



426 USE OF TABLE III. 

different aspect, in order that it may be subservient to the con- 
struction of catenarian arches of equilibration, whether for 
bridges or for powder magazines. 

It is very well known with regard to arches of equilibration, 
when the substance of the structure presses vertically down- 
wards, by the force of gravity, that for a parabolic arch as- 
sumed as the intrados, the principles of equiponderance require 
a similar and equal parabolic curve, situated throughout at the 
same vertical distance from the inner curve as the extrados ; 
and in that case the curves may be easily constructed and the 
joints of the voussoirs found, by the methods explained in p. 
165 and 167. But it is equally true, that upon the same general 
principles of equipollence a catenarian curve may be assumed 
for the arch, and the intrados and extrados be two similar and 
equidistant curves, provided that equidistance be measured upon 
the radius of curvature at each point, and be but small compared 
with the span of the arch. Constructions founded upon the 
knowledge of this fact have been long rejected, but have of late 
been re-introduced ; on which account Table III. is given, that 
the time of practical men may not be wasted in needless calcu- 
lation. 

Suppose that the marginal figure in the lower part of p. 164 
represented a catenary, in which, as indicated in the table, a p 
represented any portion of the curve a g, and g p, the corres- 
ponding abscissa and ordinate, and g k, the subnormal, p repre- 
senting the parameter, or constant horizontal tension at the ver- 
tex of the curve. Then, in addition to the equations exhibited 
in p. 175, &c, it is known that ap= ^ (a g) 2 + 2jo a g. 

an d I =- 3AG 2 (AG) 3 26 (a ■ g)' 662 (a g) 7 

p ' " (g p) 3 3 (g p) 4 > 45 (g p) 6 945 (g p) 8 "*" ' 

Now, the table presents values of a g, g p, g k, correspond- 
ing to several values of a p, the arch, on the supposition that p, 
the parameter, or tension at the vertex, is 25. 

Suppose, as an illustration of the use of the table, it were re- 
quired to construct a vault whose semibase should be 8 (yards, 
for example) and the height 10 ; we shall find its model by 
searching that part of the table where the abscissa, a g, and the 
ordinate, g p, are in the ratio of 10 to 8. This is, where the arc 
a p is 80 ; for there the abscissa and ordinate are 58*81 and 
46-99, which are in the ratio of 10*01 to 8, sufficiently near for 
practical purposes. Hence, then, the arch may be regarded as 
divided into 80 equal parts, and the table will present the com- 
puted proportions for 80 voussoirs, on each side of the vertex, 
or rather for 79 voussoirs and half the key stone. Or they may 
be reduced to 40, or to 20, or to 10, on each side the crown of 



USE OF TABLE III. 427 

the structure. The actual values to the dimensions 10 and 8, 

will be found by multiplying each number in the table by ? 

or its equivalent 0-1702 ; or, to have the dimensions in feet, let 
the multiplier be 3 x 0*1702 or 0-5106. Thus, we should have 
the values of a g, g p, a p, for every voussoir from the vertex 
of the curve downwards, while the corresponding values of 
a g + g k would give the points k, from which the line k p 
must be drawn to give in each case the direction of the joint. 
In the example assumed, the value of the parameter, p, would 
be 25 X 0-1702 or 4*255 ; and this would be the measure of the 
horizontal thrust. 

The curve thus sketched will be posited in the middle of the 
arch, half way between the intrados and the extrados. Trace 
above and below this, at the distance \ t (half the thickness at 
the crown), measured upon the respective positions of the joints, 
two curves parallel to the mean catenary ; so shall you obtain 
the proposed arch of equilibration. 

In cases where the proposed ratio of the height and semi-span 
of the arch, cannot be found with sufficient accuracy in the 
tables, it may be approximated to by the usual methods em- 
ployed in reference to proportional parts. As, suppose the 
semi-span and the altitude were to be equal ; this condition lies 
between 60 and 61 in the values of the arch ; and, by the me- 
thod just alluded to, we shall find that when the arc is equal to 
60*44, the height and half base are each 40*40, the horizontal 
thrust being 25. 

If an equilibrated circular arch were to be erected, of the same 
span and height, and the same thickness at the crown, the hori- 
zontal thrust would be 31*6. Whence, by the way, it appears 
that an equilibrated catenarian arch in these proportions, would 
produce a less horizontal thrust than an equilibrated circular 
arch : contrary to the opinion of Bossut, Equilibres des Voutts. 



THE END. 



Plat*] 










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Ma. 3. 



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Plate TTI". 




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LIBf »ARY OF CONGRESS 




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